CH 2 SUM: Functions

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What is the inverse function of $f(x) = 2x + 3$?

$f^{-1}(x) = 2x - 3$

$f^{-1}(x) = \frac{x}{2} + 3$

$f^{-1}(x) = \frac{x - 3}{2}$ (correct)

$f^{-1}(x) = \frac{x + 3}{2}$

Which of the following is the general form of an exponential function?

$y = x^a$, where $a > 0$ and $a \neq 1$

$y = a^x$, where $a < 0$ and $a \neq 1$

$y = a^x$, where $a > 0$ and $a \neq 1$ (correct)

$y = a^x$, where $a$ is any real number

What is the inverse function of $f(x) = 3^x$?

$f^{-1}(x) = x^{\log_3(x)}$

$f^{-1}(x) = 3^{\log_x(3)}$

$f^{-1}(x) = \log_3(x)$ (correct)

$f^{-1}(x) = \log_x(3)$

Which condition must be satisfied for a quadratic function to have an inverse?

The domain must be restricted to $x \geq 0$. Signup and view all the answers

What is the graphical representation of a linear function with a non-zero slope?

A straight line Signup and view all the answers

Which type of function is a one-to-one function?

A linear function with a non-zero slope Signup and view all the answers

What is the graphical representation of an exponential function with a base greater than 1?

A rapidly increasing curve Signup and view all the answers

Which test determines if a graph represents a function?

The vertical line test Signup and view all the answers

What is the condition for a function to have an inverse function?

The function must be one-to-one (injective). Signup and view all the answers

Which of the following is an example of a many-to-one function?

A quadratic function Signup and view all the answers

What is the process for deriving the inverse of a linear function $y = ax + q$?

<ol> <li>Interchange x and y. 2. Isolate y.</li> </ol> Signup and view all the answers

If $f(x) = -3x + 1$, what is the inverse function $f^{-1}(x)$?

$(-1/3)x + 1/3$ Signup and view all the answers

What is the graphical relationship between a function and its inverse?

They are reflected across the line y = x. Signup and view all the answers

What happens to the domain and range when a linear function is inverted?

The domain and range are swapped. Signup and view all the answers

In the standard form of a quadratic function $y = ax^2 + bx + c$, what does the coefficient $a$ determine?

The opening direction and width of the parabola Signup and view all the answers

What is the axis of symmetry for a quadratic function?

The line x = -b/2a Signup and view all the answers

What are the asymptotes of a hyperbolic function $y = a/(x - h) + k$?

Vertical asymptote: x = h, Horizontal asymptote: y = k Signup and view all the answers

What determines the orientation of a hyperbolic function?

The coefficient a and the shifts h and k Signup and view all the answers

What is the domain of a quadratic function?

All real numbers Signup and view all the answers

How is the range of a hyperbolic function determined?

By the positions of the asymptotes Signup and view all the answers

What is the defining characteristic of a function?

Every element in the domain is associated with exactly one element in the range Signup and view all the answers

What is the defining characteristic of an inverse function?

The inverse function maps from the range of the original function to the domain Signup and view all the answers

What is the requirement for the existence of an inverse function?

The original function must be bijective (both one-to-one and onto) Signup and view all the answers

What is the general form of a linear function?

$y = mx + c$ Signup and view all the answers

Which of the following is NOT a fundamental concept in mathematical analysis?

Derivatives Signup and view all the answers

What is the relationship between a function and its inverse function?

They reverse the assignment made by the original function Signup and view all the answers

What is the inverse of a linear function expressed as $f(x) = 4x - 2$?

$f^{-1}(x) = \frac{x - 2}{4}$ Signup and view all the answers

Which test is used to determine if a function has an inverse that is also a function?

Horizontal Line Test Signup and view all the answers

For the function $f(x) = 5^x$, what is the inverse function?

$f^{-1}(x) = \log_5(x)$ Signup and view all the answers

What is the inverse of a quadratic function $f(x) = -3x^2$, considering $x$ non-negative?

$f^{-1}(x) = -\sqrt{x/3}$ Signup and view all the answers

In a linear function $f(x) = -2x + 5$, what does the value of '-2' represent?

The slope of the line Signup and view all the answers

What does the y-intercept signify in a linear function's graph?

The point where the line intersects the y-axis Signup and view all the answers

For a linear function in the form $y = 3x - 7$, if you set $y$ to zero to find an intercept, which intercept are you calculating?

X-intercept Signup and view all the answers

What is the domain and range of linear functions?

$Domain: \mathbb{R}, Range: \mathbb{R}$ Signup and view all the answers

What does a positive slope 'a' indicate in a linear function?

An upward trajectory of the line Signup and view all the answers

How is the gradient (slope) calculated in a linear function?

$\text{Gradient} = \frac{\text{change in y}}{\text{change in x}}$ Signup and view all the answers

What is the typical horizontal asymptote of an exponential function?

The x-axis Signup and view all the answers

Which logarithmic law states that $\log_b(x^p) = p \cdot \log_b(x)$?

Power Rule Signup and view all the answers

How does the base $b$ of an exponential function $f(x) = b^x$ affect the function's behavior?

The base determines whether the function exhibits growth or decay Signup and view all the answers

What is the relationship between exponential and logarithmic functions?

Logarithmic functions are the inverse of exponential functions Signup and view all the answers

Which transformation can be applied to both exponential and logarithmic graphs?

All of the above Signup and view all the answers

How can exponential equations be solved?

Both a and b Signup and view all the answers

What is the domain of a logarithmic function $y = \log_b(x)$?

Positive real numbers ($0 < x < \infty$) Signup and view all the answers

What is the y-intercept of an exponential function $f(x) = b^x$?

The y-intercept is always at (0, 1) Signup and view all the answers

Which logarithmic law states that $\log_b(xy) = \log_b(x) + \log_b(y)$?

Product Rule Signup and view all the answers

What is the typical vertical asymptote of a logarithmic function $y = \log_b(x)$?

The y-axis Signup and view all the answers

What is the defining characteristic of an inverse function?

Maps from the range back to the domain Signup and view all the answers

Which condition must be met for an inverse function to exist?

Bijective Signup and view all the answers

In a linear function $f(x) = 4x - 2$, what does the value '4' represent?

Slope coefficient Signup and view all the answers

What is the role of a linear function's y-intercept?

Indicates a point of intersection with the y-axis Signup and view all the answers

Which type of function must a linear function be to have an inverse?

Bijective Signup and view all the answers

How is an inverse function symbolically denoted?

$f^{-1}$ Signup and view all the answers

What is the key characteristic that determines the direction of a parabolic function?

The coefficient $a$ Signup and view all the answers

Which of the following is NOT a characteristic of the inverse of a linear function?

The function remains linear Signup and view all the answers

What is the relationship between the vertex of a quadratic function and the axis of symmetry?

The vertex is located on the axis of symmetry Signup and view all the answers

What is the defining characteristic of a hyperbolic function?

The graph has two distinct branches located in opposite quadrants Signup and view all the answers

How does the domain and range of a quadratic function compare to the domain and range of its inverse?

The domain and range are interchanged Signup and view all the answers

What is the relationship between the $y$-intercept of a linear function and the $x$-intercept of its inverse?

The $y$-intercept of the original function becomes the $x$-intercept of the inverse Signup and view all the answers

What is the relationship between the $x$-intercept of a linear function and the $y$-intercept of its inverse?

The $x$-intercept of the original function becomes the $y$-intercept of the inverse Signup and view all the answers

What is the general form of a hyperbolic function?

$y = a/(x - h) + k$ Signup and view all the answers

Which of the following is NOT a characteristic of the axis of symmetry in a quadratic function?

It is the line where the parabola changes direction Signup and view all the answers

What is the relationship between the domain and range of a hyperbolic function and its asymptotes?

The domain and range are determined by the position of the asymptotes Signup and view all the answers

What is the inverse of the quadratic function $f(x) = 4x^2$ when considering $x$ to be non-negative?

$f^{-1}(x) = 2rac{1}{ oot 4 ext{x}}$ Signup and view all the answers

For a linear function in the form $y = -3x + 2$, what role does the value '-3' play?

It signifies the function's slope Signup and view all the answers

In a linear function $y = 4x - 7$, setting $y$ to zero to find an intercept determines which intercept?

X-intercept Signup and view all the answers

What does the Horizontal Line Test confirm regarding a function's graph?

One-to-one nature of the function Signup and view all the answers

Which characteristic does a negative slope in a linear function signify?

Increasing values on the x-axis lead to decreasing values on the y-axis Signup and view all the answers

What is the inverse of an exponential function $f(x) = e^x$?

$f^{-1}(x) = ext{ln}(x)$ Signup and view all the answers

For a linear function $f(x) = mx + c$, what does the value of $m$ represent?

The slope Signup and view all the answers

What is the inverse function of $f(x) = 2x^2 + 3$?

$f^{-1}(x) = \pm \sqrt{\frac{x - 3}{2}}$ Signup and view all the answers

Which test is used to determine if a function is one-to-one (injective)?

Vertical Line Test Signup and view all the answers

For the exponential function $f(x) = 2^x$, what is the inverse function?

$f^{-1}(x) = \log_2(x)$ Signup and view all the answers

What is the condition for a function to have an inverse that is also a function?

The function must be one-to-one and onto Signup and view all the answers

What is the graphical representation of the inverse of a function?

The same graph, but reflected across the line $y = x$ Signup and view all the answers

For a quadratic function $f(x) = ax^2 + bx + c$, what does the coefficient $a$ determine?

The orientation of the parabola Signup and view all the answers

Which type of function is a one-to-one function?

Linear functions with non-zero slope Signup and view all the answers

What is the horizontal asymptote of the exponential function $f(x) = 2^x$?

$y = 0$ Signup and view all the answers

What is the inverse function of $f(x) = \log_3(x)$?

$f^{-1}(x) = 3^x$ Signup and view all the answers

Which step is crucial when graphing quadratic functions?

Finding the vertex coordinates Signup and view all the answers

What is the significance of the base $b$ in an exponential function $f(x) = b^x$?

It affects the function's growth or decay behavior Signup and view all the answers

How can the intersection point between two functions be determined?

By equating the two functions and solving for the variable Signup and view all the answers

What is the impact of reflecting a function across the line $y = x$?

All of the above Signup and view all the answers

Which transformation can be applied to both exponential and logarithmic functions?

All of the above Signup and view all the answers

What is the range of a logarithmic function $y = \log_b(x)$?

All real numbers $\mathbb{R}$ Signup and view all the answers

What is the inverse function of $f(x) = 2^x$?

$f^{-1}(x) = \log_2(x)$ Signup and view all the answers

How can exponential equations be solved?

By isolating the exponential expression and applying logarithms to both sides Signup and view all the answers

What is the typical vertical asymptote of a logarithmic function $y = \log_b(x)$?

$x = 0$ Signup and view all the answers

Which condition must be satisfied for a quadratic function to have an inverse?

The function must be one-to-one Signup and view all the answers

If a function $f(x)$ is bijective, what can be said about its inverse function $f^{-1}(x)$?

$f^{-1}(x)$ is also a bijective function. Signup and view all the answers

Given a linear function $f(x) = mx + c$, what is the relationship between the slope $m$ and the inverse function $f^{-1}(x)$?

The inverse function $f^{-1}(x)$ has a slope of $1/m$. Signup and view all the answers

For a quadratic function $f(x) = ax^2 + bx + c$, what condition must be satisfied for it to have an inverse function?

$a \neq 0$ and $b = 0$ Signup and view all the answers

If $f(x) = \log_b(x)$ and $g(x) = b^x$, what is the relationship between $f$ and $g$?

$f(x)$ is the inverse function of $g(x)$. Signup and view all the answers

For a hyperbolic function $f(x) = \frac{a}{x - h} + k$, what is the condition for it to have a vertical asymptote?

$a \neq 0$ and $h \neq 0$ Signup and view all the answers

If $f(x) = \sqrt{x}$ and $g(x) = x^2$, which of the following statements is true?

$f(x)$ is the inverse function of $g(x)$. Signup and view all the answers

What impact does the base have on an exponential function when $b > 1$?

Exponential growth Signup and view all the answers

For a linear function in the form $f(x) = 4x - 2$, what role does the value '4' play?

Determines the direction of the line Signup and view all the answers

What would be the inverse function of $f(x) = rac{1}{5}x + 2$?

$f^{-1}(x) = 5x - 10$ Signup and view all the answers

What is the range of an exponential function $f(x) = b^x$?

All positive real numbers Signup and view all the answers

What does the y-intercept signify in a linear function's graph?

Specific location on the y-axis Signup and view all the answers

In a logarithmic function $y = \log_b(x)$, what occurs at $x = 0$?

Vertical asymptote Signup and view all the answers

In a linear function $f(x) = -3x + 2$, what does the value '-3' represent?

Determines the direction of the line Signup and view all the answers

Which property helps in converting logarithmic functions to exponential forms for easier calculations?

Change of base formula Signup and view all the answers

How can a linear function's graph be constructed using the Gradient-Intercept approach?

By starting at the y-intercept and using slope to plot another point Signup and view all the answers

What is the primary focus when sketching the graph of an exponential function?

Identifying the y-intercept Signup and view all the answers

What is the common outcome when reflecting a function across the line $y = x$?

$f(x)$ becomes its inverse Signup and view all the answers

What is significant about the domain and range of linear functions?

The domain and range extend over all real numbers Signup and view all the answers

What is the primary reason for restricting the domain in quadratic functions?

To create a one-to-one relationship for an inverse function Signup and view all the answers

$rac{\log_k(a)}{\log_k(b)}$ is used in which logarithmic formula?

$y = \log_b(x)$ Signup and view all the answers

What is an essential condition for a linear function to have an inverse?

The function must pass the Vertical Line Test Signup and view all the answers

What does setting $y = 0$ help determine in a logarithmic function?

$x$-intercept Signup and view all the answers

$f(x) = b^x$ depicts exponential growth when which condition is met?

$b > 1$ Signup and view all the answers

What role does the gradient ('a') play in determining a linear function's orientation?

Determines if the line is upward or downward-sloping Signup and view all the answers

How does restricting a quadratic function's domain lead to one-to-one relationships?

By preventing multiple inputs from producing the same output Signup and view all the answers

$f(x) = \log_b(x)$ intersects the x-axis at which point?

(1, 0) Signup and view all the answers

What role does the line $y = x$ play in representing the relationship between a linear function and its inverse?

It acts as the mirroring axis for the functions Signup and view all the answers

In the standard form of a quadratic function $y = ax^2 + bx + c$, what does the coefficient $b$ determine?

The roots or zeroes of the function Signup and view all the answers

What aspect of a hyperbolic function's graph is governed by the shifts indicated by $h$ and $k$ in its general form?

The orientation Signup and view all the answers

Which characteristic differentiates the roles of positive and negative values of parameter $a$ in a quadratic function?

Defining the direction of the parabola Signup and view all the answers

When graphing a quadratic function, what is the significance of computing the y-intercept by setting $x = 0$?

Identifying a crucial point on the graph Signup and view all the answers

In a linear function $f(x) = -2x + 4$, what is the significance of the value '-2'?

It defines the direction of the function Signup and view all the answers

What is an essential step when constructing the graph of a hyperbolic function with general form $y = \frac{a}{x - h} + k$?

Computing horizontal and vertical asymptotes Signup and view all the answers

What characteristic is indicated by a negative slope 'a' in a linear function?

Downward orientation of the function Signup and view all the answers

What component dictates whether a parabolic function opens upward or downward?

The parameter 'a' Signup and view all the answers

When considering a quadratic function in standard form $y=ax^2+bx+c$, what important information is derived from finding its roots or zeroes?

The x-intercepts Signup and view all the answers

What is the key property that must be satisfied for a function to have an inverse function?

The function must be one-to-one (bijective) Signup and view all the answers

How does the domain and range of a quadratic function compare to the domain and range of its inverse function?

The domain and range of the inverse function are the same as the range and domain of the quadratic function, respectively Signup and view all the answers

What is the inverse function of $f(x) = 3^x$?

$f^{-1}(x) = \"log_3(x)$ Signup and view all the answers

What is the graphical relationship between a function and its inverse function?

The graphs are reflections of each other across the line $y = x$ Signup and view all the answers

Which test is used to determine if a function has an inverse that is also a function?

The Horizontal Line Test Signup and view all the answers

What is the inverse function of $f(x) = 2x^2 + 3$, considering $x$ non-negative?

$f^{-1}(x) = \"sqrt{(x - 3)/2}$ Signup and view all the answers

What is the typical horizontal asymptote of an exponential function $f(x) = b^x$?

The horizontal asymptote is $y = 0$ Signup and view all the answers

How does the base $b$ of an exponential function $f(x) = b^x$ affect the function's behavior?

If $b > 1$, the function increases exponentially; if $0 Signup and view all the answers

What is the inverse function of $f(x) = -3x^2$, considering $x$ non-negative?

$f^{-1}(x) = \"sqrt{x/3}$ Signup and view all the answers

What is the defining characteristic of an inverse function?

An inverse function reverses the assignments made by the original function Signup and view all the answers

If the function $f(x) = 3x + 5$ is invertible, what is the inverse function $f^{-1}(x)$?

$\frac{x - 5}{3}$ Signup and view all the answers

What is the condition for an exponential function $f(x) = b^x$ to be one-to-one?

$b > 0$ and $b \neq 1$ Signup and view all the answers

For the quadratic function $f(x) = x^2 - 4x + 3$, what is the inverse function $f^{-1}(x)$ on the domain $x \geq 3$?

$\pm\sqrt{x + 3} - 2$ Signup and view all the answers

If $f(x) = \log_2(x)$ and $g(x) = 2^x$, what is the relationship between $f$ and $g$?

$g(x)$ is the inverse of $f(x)$ Signup and view all the answers

For a bijective function $f: \mathbb{R} \rightarrow \mathbb{R}$, which statement is true about its inverse $f^{-1}$?

The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$ Signup and view all the answers

What is the inverse function of $f(x) = \frac{1}{2}x + 3$?

$g(x) = \frac{x - 3}{2}$ Signup and view all the answers

Which of the following is the primary requirement for a function to have an inverse function?

The function must be bijective Signup and view all the answers

What is the relationship between the domain of a function and the range of its inverse function?

The domain of the function becomes the range of the inverse function Signup and view all the answers

What is the graphical representation of the inverse of a linear function $y = mx + c$?

A line that is the reflection of the original function across the line $y = x$ Signup and view all the answers

Which of the following is a characteristic of an inverse function?

It reverses the assignment made by the original function Signup and view all the answers

What is the relationship between the slope of a linear function and the slope of its inverse function?

The slope of the inverse function is the reciprocal of the slope of the original function Signup and view all the answers

What is the defining characteristic of a bijective function?

It is a one-to-one function Signup and view all the answers

What is the key property that must be satisfied for a function to have an inverse function?

The function must be bijective (one-to-one and onto) Signup and view all the answers

Which of the following is the correct formula for the inverse of a linear function $f(x) = mx + c$?

$f^{-1}(x) = \frac{x - c}{m}$ Signup and view all the answers

What is the inverse function of the exponential function $f(x) = 3^x$?

$f^{-1}(x) = \log_3(x)$ Signup and view all the answers

What is the role of the horizontal line test in determining whether a function has an inverse function?

The horizontal line test determines if the function is one-to-one Signup and view all the answers

Which of the following statements is true about the graph of an exponential function $f(x) = b^x$ where $b > 1$?

The graph is a rapidly increasing curve that approaches the x-axis as an asymptote Signup and view all the answers

What is the relationship between the vertex of a quadratic function $f(x) = ax^2 + bx + c$ and the axis of symmetry?

The vertex is located at the point where the axis of symmetry intersects the parabola Signup and view all the answers

What is the impact of reflecting a function $f(x)$ across the line $y = x$?

The reflected function becomes $f^{-1}(x) Signup and view all the answers

What is the significance of the y-intercept in the graph of a linear function $f(x) = mx + b$?

The y-intercept represents the value of the function when $x = 0$ Signup and view all the answers

What is the impact of the coefficient $a$ in the standard form of a quadratic function $f(x) = ax^2 + bx + c$?

The coefficient $a$ determines the direction of opening of the parabola (upward or downward) Signup and view all the answers

Which of the following functions is an example of a many-to-one function?

The quadratic function $f(x) = x^2$ Signup and view all the answers

What is the graphical representation of a linear function with a non-zero slope?

A straight line Signup and view all the answers

For a quadratic function $f(x) = ax^2 + bx + c$, what does the coefficient $a$ determine?

The orientation of the parabola Signup and view all the answers

What is the condition for a function to have an inverse that is also a function?

The function must be bijective (one-to-one and onto) Signup and view all the answers

Which test is used to determine if a function has an inverse that is also a function?

Horizontal Line Test Signup and view all the answers

What is the inverse function of $f(x) = 3^x$?

$f^{-1}(x) = \log_3(x)$ Signup and view all the answers

What is the relationship between exponential and logarithmic functions?

They are inverses of each other Signup and view all the answers

In a linear function $f(x) = mx + c$, what does the value $m$ represent?

The slope Signup and view all the answers

What is the typical horizontal asymptote of an exponential function $f(x) = b^x$?

The x-axis Signup and view all the answers

What is the axis of symmetry for a quadratic function?

The line $x = \frac{-b}{2a}$ Signup and view all the answers

What is the defining characteristic of a hyperbolic function?

It has two asymptotes Signup and view all the answers

What is the graphical relationship between a function and its inverse?

The graphs are reflections of each other across the line y = x Signup and view all the answers

For the linear function $f(x) = mx + c$, what is the inverse function?

$f^{-1}(x) = \frac{x - c}{m}$ (given $m \neq 0$) Signup and view all the answers

What is the inverse function of $f(x) = a^x$, where $a$ is a positive constant?

$f^{-1}(x) = \log_a(x)$ Signup and view all the answers

For a quadratic function $f(x) = ax^2$ (with $a > 0$ and $x \geq 0$), what is the inverse function?

$f^{-1}(x) = \sqrt{\frac{x}{a}}$ Signup and view all the answers

What test is used to determine if a function is injective (one-to-one)?

Horizontal Line Test Signup and view all the answers

What is the primary condition for a function to have an inverse that is also a function?

The function must be injective (one-to-one) Signup and view all the answers

If $f(x) = \log_3(x)$, what is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = 3^x$ Signup and view all the answers

For a quadratic function $f(x) = 2x^2$ with $x \geq 0$, what is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \sqrt{\frac{x}{2}}$ Signup and view all the answers

What is the inverse function of $f(x) = 2x + 3$?

$f^{-1}(x) = \frac{x - 3}{2}$ Signup and view all the answers

What is the inverse function of $f(x) = \frac{1}{5}x + 2$?

$f^{-1}(x) = \frac{x - 2}{5}$ Signup and view all the answers

What is the inverse function of $y = 2x + 3$?

$x = y/2 - 3/2$ Signup and view all the answers

If $f(x) = x^2 - 4x + 3$, what is the vertex form of this quadratic function?

$y = -(x - 2)^2 + 3$ Signup and view all the answers

For a hyperbolic function $y = \frac{3}{x + 2} - 1$, what is the horizontal asymptote?

$y = -1$ Signup and view all the answers

What is the axis of symmetry for the quadratic function $y = -2x^2 + 4x - 3$?

$x = 1$ Signup and view all the answers

For a linear function $f(x) = mx + c$, what happens to the slope $m$ in the inverse function $f^{-1}(x)$?

The slope becomes $1/m$ Signup and view all the answers

If $f(x) = 3^x$, what is the inverse function $f^{-1}(x)$?

$\log_3(x)$ Signup and view all the answers

For a quadratic function $y = ax^2 + bx + c$, what does the value of $c$ represent?

The y-intercept Signup and view all the answers

What is a necessary condition for a linear function $f(x)$ to have an inverse function $f^{-1}(x)$?

$f(x)$ must be one-to-one Signup and view all the answers

What is the inverse function of $y = \log_2(x)$?

$x = 2^y$ Signup and view all the answers

For the function $f(x) = \frac{1}{x - 2} + 4$, what is the vertical asymptote?

$x = 2$ Signup and view all the answers

In the exponential function $f(x) = b^x$, what happens when the base $b$ is between 0 and 1?

The function exhibits exponential decay Signup and view all the answers

What is the range of the logarithmic function $y = \log_b(x)$?

All real numbers $\mathbb{R}$ Signup and view all the answers

Which logarithmic law states that $\log_b(x^p) = p \cdot \log_b(x)$?

Power Rule Signup and view all the answers

How can exponential equations be solved?

By isolating the exponential expression and applying logarithms to both sides Signup and view all the answers

What is the typical horizontal asymptote of an exponential function?

The line $y = 0$ (x-axis) Signup and view all the answers

What transformation can be applied to both exponential and logarithmic functions?

All of the above Signup and view all the answers

If $f(x) = \log_b(x)$ and $g(x) = b^x$, what is the relationship between $f$ and $g$?

$g$ is the inverse of $f$ Signup and view all the answers

What is an essential step when constructing the graph of a hyperbolic function with general form $y = \frac{a}{x - h} + k$?

Identify the horizontal and vertical asymptotes Signup and view all the answers

In the standard form of a quadratic function $y = ax^2 + bx + c$, what does the coefficient $a$ determine?

The orientation (opening upward or downward) Signup and view all the answers

What is the inverse function of $f(x) = 2^x$?

$f^{-1}(x) = \log_2(x)$ Signup and view all the answers

Which of the following is NOT a key characteristic of linear functions?

The function is constrained to pass through a single point Signup and view all the answers

What is the purpose of the Horizontal Line Test (HLT) when analyzing functions?

To determine if a function has an inverse that is also a function Signup and view all the answers

Suppose $f(x) = 3^x$. What is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \log_3(x)$ Signup and view all the answers

For a linear function $f(x) = 2x + 3$, what is the expression for the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \frac{x - 3}{2}$ Signup and view all the answers

What is the graphical representation of the inverse of a linear function?

The graph is a reflection of the original function across the line $y = x$ Signup and view all the answers

Which of the following is a necessary condition for a function to have an inverse function?

The function must be injective (one-to-one) Signup and view all the answers

What is the inverse of the quadratic function $f(x) = 2x^2$ when the domain is restricted to non-negative values of $x$?

$f^{-1}(x) = \sqrt{x/2}$ Signup and view all the answers

What is the role of the slope 'a' in a linear function $f(x) = ax + c$?

It determines the angle and direction of the line Signup and view all the answers

Which logarithmic property is used to express $\log_b(x^p)$ in terms of $\log_b(x)$?

$\log_b(x^p) = p \cdot \log_b(x)$ Signup and view all the answers

What is the domain of a logarithmic function $y = \log_b(x)$?

The domain is all real numbers $x > 0$ Signup and view all the answers

When graphing a hyperbolic function $f(x) = \frac{a}{x - h} + k$, which of the following steps is crucial in determining the asymptotes?

Ascertain the horizontal and vertical asymptotes from $h$ and $k$. Signup and view all the answers

For a quadratic function in vertex form $y = a(x - h)^2 + k$, what does the parameter $a$ represent?

The width and opening direction of the parabola. Signup and view all the answers

What is the relationship between a linear function $f(x) = mx + c$ and the slope of its inverse function $f^{-1}(x)$?

The slope of $f^{-1}(x)$ is the reciprocal of $m$. Signup and view all the answers

In the logarithmic function $y = \log_b(x)$, what is the significance of the base $b$?

It represents the rate of growth or decay. Signup and view all the answers

Which condition must be satisfied for a quadratic function $f(x) = ax^2 + bx + c$ to have an inverse that is also a function?

The discriminant $b^2 - 4ac$ must be positive. Signup and view all the answers

In the exponential function $f(x) = b^x$, how does the base $b$ affect the function's behavior?

If $b > 1$, the function exhibits exponential growth; if $0 Signup and view all the answers

What is the graphical representation of the inverse of a function $f(x)$?

The graph of $f(x)$ reflected across the line $y = x$. Signup and view all the answers

In a quadratic function $f(x) = ax^2 + bx + c$, what does the value of $b$ represent?

The slope of the axis of symmetry. Signup and view all the answers

Which logarithmic law states that $\log_b(x^p) = p \cdot \log_b(x)$?

Power rule Signup and view all the answers

What is the significance of the y-intercept in a linear function $f(x) = mx + c$?

It is the point where the line crosses the y-axis. Signup and view all the answers

What is the key difference between exponential growth and decay functions?

The sign of the base Signup and view all the answers

In an exponential function, if the base is a negative number, what impact does it have on the function's graph?

It reflects the graph across the x-axis Signup and view all the answers

What is the primary reason for requiring $b > 0$ in the base of an exponential function?

To guarantee the function's y-intercept at (0, 1) Signup and view all the answers

Which property is unique to logarithmic functions compared to exponential functions?

Intersecting the x-axis at (1, 0) Signup and view all the answers

What does the 'Change of Base Formula' help accomplish in logarithmic functions?

Simplifies solving logarithmic equations Signup and view all the answers

How do logarithmic functions compare to exponential functions in terms of their inverse relationship?

Logarithmic functions are inverses of exponential functions Signup and view all the answers

What distinguishes the domain of an exponential function from that of a logarithmic function?

Exponential functions cover all real numbers while logarithmic functions exclude certain values Signup and view all the answers

'Stretches and compressions' are transformations that primarily affect which aspect of exponential and logarithmic graphs?

'Stretches and compressions' modify the steepness or width of the graph Signup and view all the answers

'Inverse Relationship' is a core property shared by which pair of mathematical functions?

'Exponential' and 'Logarithmic' functions Signup and view all the answers

What is the primary condition for a function to have an inverse that is also a function?

The function must be bijective (one-to-one and onto) Signup and view all the answers

What is the graphical relationship between a function and its inverse function?

The graphs are reflections of each other across the line $y = x$ Signup and view all the answers

If $f(x) = \log_b(x)$ and $g(x) = b^x$, what is the relationship between $f$ and $g$?

$f(x) = g^{-1}(x)$ Signup and view all the answers

What is the inverse function of $f(x) = 3^x$?

$f^{-1}(x) = \log_3(x)$ Signup and view all the answers

For a linear function $f(x) = mx + c$, what is the relationship between the $y$-intercept of $f(x)$ and the $x$-intercept of its inverse function $f^{-1}(x)$?

The $y$-intercept of $f(x)$ is the $x$-intercept of $f^{-1}(x)$ Signup and view all the answers

What is the typical horizontal asymptote of an exponential function $f(x) = b^x$?

$y = 0$ Signup and view all the answers

What is the fundamental operation that inverses of linear functions perform?

Subtraction Signup and view all the answers

When deriving the inverse of a linear function, what step involves isolating y in the equation?

Transitioning from y = ax + q to x = ay + q Signup and view all the answers

In the inverse of a linear function, how do the original function's y-intercept and the inverse's x-intercept relate?

They switch positions Signup and view all the answers

What property signifies the symmetry between a linear function and its inverse on a graph?

Symmetry axis y = x Signup and view all the answers

In a quadratic function, what does 'a' represent in the standard form y = ax^2 + bx + c?

Determines parabola's width Signup and view all the answers

What characteristic of a quadratic function is indicated by the vertex point?

Peak or trough location Signup and view all the answers

For a quadratic function, how are x-intercepts typically found?

By factoring or using the quadratic formula Signup and view all the answers

What aspect of a hyperbolic function is determined by the position of its asymptotes?

Domain and Range Signup and view all the answers

In hyperbolic functions, what is meant by 'orientation' as governed by the coefficient 'a'?

The direction of opening of each branch Signup and view all the answers

When graphing a parabolic function, what is calculated by setting x = 0?

The y-intercept Signup and view all the answers

What is the inverse of a linear function $f(x) = -4x + 2$?

$f^{-1}(x) = -\frac{x - 2}{4}$ Signup and view all the answers

For a quadratic function $f(x) = -x^2 + 3x + 1$, what does the coefficient $-1$ represent?

Vertex of the parabola Signup and view all the answers

Which type of function is most likely to violate the one-to-one property?

Quadratic function Signup and view all the answers

What is the key condition for a function to have an inverse that is also a function?

It must pass the horizontal line test Signup and view all the answers

What is the primary role of function notation like $f(x)$ and $f^{-1}(x)$?

To indicate the value of the original and inverse functions at a given point Signup and view all the answers

In an exponential function $f(x) = rac{1}{2}^x$, what impact does having a base less than 1 have on the graph?

The graph shifts downwards Signup and view all the answers

What does bijectivity imply when considering a function and its inverse?

Every element in the domain maps to exactly one element in the range Signup and view all the answers

$f(x) = -rac{1}{3}x + 4$ is an example of which type of function?

Linear function Signup and view all the answers

$f(x) = 5^x$ depicts exponential growth when which condition is met?

$f(x)$ is always positive for all real $x$ Signup and view all the answers

$f(x) = rac{3}{x - 1} + 2$ represents what type of function?

Logarithmic function Signup and view all the answers

What does the base 'b' determine in an exponential function of the form $f(x) = b^x$?

The growth or decay behavior Signup and view all the answers

Which of the following is true about the asymptote of a logarithmic function?

It has a vertical orientation Signup and view all the answers

What is the key difference between the domain of exponential functions and logarithmic functions?

Exponential functions include all real numbers in their domain Signup and view all the answers

Which of the following is a core attribute of logarithmic functions?

Being the inverse of exponential functions Signup and view all the answers

What is the primary property that distinguishes exponential growth from exponential decay?

The base value 'b' Signup and view all the answers

Which transformation technique applies to both exponential and logarithmic graphs to alter their shapes?

Vertical stretching Signup and view all the answers

What is the primary difference in behavior between exponential functions with different base values 'b'?

'b' influences the function's growth or decay Signup and view all the answers

When graphing logarithmic functions, what key point defines their behavior near x = 0?

(0, 1) Signup and view all the answers

If a quadratic function $f(x) = x^2 + 2x + 3$ satisfies the horizontal line test, what can be said about its inverse?

The inverse function exists and is a square root function Signup and view all the answers

In the exponential function $f(x) = 2^x$, what is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \log_2(x)$ Signup and view all the answers

What is the graphical relationship between a function $f(x)$ and its inverse $f^{-1}(x)$?

Their graphs are reflections across the line $y = x Signup and view all the answers

For a linear function $f(x) = mx + c$, if $m = 0$, what can be inferred about its inverse?

The inverse function is a constant function Signup and view all the answers

What is the primary condition for a function to have an inverse that is also a function?

The function must be injective (one-to-one) Signup and view all the answers

If $f(x) = \sqrt{x}$ and $g(x) = x^2$, what is the relationship between $f$ and $g$?

$g$ is the inverse of $f, but $f$ is not the inverse of $g Signup and view all the answers

For the linear function $f(x) = 3x - 2$, what is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \frac{x + 2}{3} Signup and view all the answers

What is the inverse function of $f(x) = \log_2(x)$?

$f^{-1}(x) = 2^x Signup and view all the answers

What is a necessary condition for an inverse function to exist?

Being bijective Signup and view all the answers

In the context of functions and inverses, what does bijectivity imply?

Unique mapping between domain and range Signup and view all the answers

For functions and their inverses, what does the value of 'm' typically represent in a linear function?

Slope Signup and view all the answers

What is a characteristic of the vertex point in the context of quadratic functions?

Minimum or maximum point on the curve Signup and view all the answers

In the context of functions, what is indicated by an exponential function having a base less than 1?

Exponential decay Signup and view all the answers

What role does the y-intercept play in the context of linear functions?

Representing the point where the curve intersects the y-axis Signup and view all the answers

Which of the following is a key characteristic of the domain and range of an exponential function $f(x) = b^x$?

The domain is all real numbers, and the range is positive real numbers. Signup and view all the answers

How can the horizontal asymptote of an exponential function $f(x) = b^x$ be determined?

The horizontal asymptote is the line $y = 0$. Signup and view all the answers

What is the relationship between an exponential function $f(x) = b^x$ and its inverse logarithmic function $g(x) = \log_b(x)$?

The functions are inverses, meaning $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$. Signup and view all the answers

How can exponential equations of the form $b^x = k$ be solved?

By converting the equation to logarithmic form and solving for $x$. Signup and view all the answers

What is the impact of reflecting a function $f(x)$ across the line $y = x$?

It interchanges the domain and range of the function. Signup and view all the answers

Which of the following is a key logarithmic law that can be used to simplify expressions involving logarithms?

All of the above Signup and view all the answers

What is the relationship between the domain and range of a hyperbolic function $f(x) = \frac{a}{x - h} + k$ and its asymptotes?

The domain is all real numbers, and the range is positive real numbers, with vertical asymptotes at $x = h$. Signup and view all the answers

What is the key property that must be satisfied for a function $f(x)$ to have an inverse function $f^{-1}(x)$?

The function must be one-to-one. Signup and view all the answers

How can the y-intercept of an exponential function $f(x) = b^x$ be determined?

The y-intercept is at the point $(0, 1)$. Signup and view all the answers

What is the impact of a negative coefficient 'a' in the standard form of a quadratic function $f(x) = ax^2 + bx + c$?

It changes the orientation of the parabola, making it open downward. Signup and view all the answers

Given the function $f(x) = 2x^2 - 3x + 1$, what restriction must be applied to the domain to ensure the existence of an inverse function?

$x \geq 0 Signup and view all the answers

If $f(x) = 4^x$ and $g(x) = \log_4(x)$, which statement is true?

$g(x)$ is the inverse of $f(x)$ Signup and view all the answers

If $f(x) = \sqrt[3]{x}$ and $g(x) = x^3$, which of the following statements is true?

Both (b) and (c) are true Signup and view all the answers

If $f(x) = \tan(x)$, which of the following statements is true about the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \arctan(x) Signup and view all the answers

If $f(x) = \log_2(x)$ and $g(x) = 2^x$, which of the following statements is true?

Both (b) and (c) are true Signup and view all the answers

If $f(x) = \frac{1}{x}$, which of the following statements is true about the inverse function $f^{-1}(x)$?

$f^{-1}(x) = x Signup and view all the answers

If $f(x) = \sqrt{x}$ and $g(x) = x^2$, which of the following statements is true?

$f(g(x)) = x$ for all $x$ in the domain of $g(x) Signup and view all the answers

If $f(x) = \log_3(x)$ and $g(x) = 3^x$, which of the following statements is true?

Both (b) and (c) are true Signup and view all the answers

If $f(x) = \arcsin(x)$ and $g(x) = \sin(x)$, which of the following statements is true?

$f(x)$ and $g(x) Signup and view all the answers

What is the primary role of the line $y = x$ in relation to linear function inverses?

Symbolizing the mirroring axis for the function and its inverse Signup and view all the answers

When finding the inverse of a linear function, which step is pivotal in isolating $y$ in the equation?

Isolating $y$ in terms of $x Signup and view all the answers

What property ensures that both linear functions and their inverses remain true linear functions?

Bijectivity Signup and view all the answers

In a linear function, if the coefficient $m$ in $f(x) = mx + c$ is negative, what does this indicate about the slope of its inverse?

Positive slope Signup and view all the answers

What critical aspect distinguishes constant relationships depicted by linear functions and their inverses?

Domain and range transitions Signup and view all the answers

Which characteristic is NOT typically associated with the axis of symmetry in quadratic functions?

Identifying the roots or zeroes Signup and view all the answers

Which of the following best describes the graphical relationship between a function $f(x)$ and its inverse $f^{-1}(x)$?

The graphs are reflections of each other across the line $y = x$. Signup and view all the answers

For a quadratic function $f(x) = ax^2$ (with $a > 0$ and $x \geq 0$), what is the inverse function $f^{-1}(x)$?

$f^{-1}(x) = \sqrt{x/a}$ Signup and view all the answers

What is the inverse function of $f(x) = 2x + 3$?

$f^{-1}(x) = \frac{x - 3}{2}$ Signup and view all the answers

Which test is used to determine if a function is one-to-one (injective)?

Horizontal Line Test Signup and view all the answers

What is the inverse function of $f(x) = 3^x$?

$f^{-1}(x) = \log_3(x)$ Signup and view all the answers

What is the primary condition for a function to have an inverse that is also a function?

The function must be one-to-one (injective). Signup and view all the answers

What is the domain of a logarithmic function $y = \log_b(x)$?

$x > 0$ Signup and view all the answers

What is the relationship between the domain of a function and the range of its inverse function?

The domain of the function becomes the range of the inverse, and vice versa. Signup and view all the answers

For a linear function $f(x) = mx + c$, what does the value of $m$ represent?

The slope of the function. Signup and view all the answers

Which of the following is NOT a key characteristic of linear functions?

The function is always one-to-one (injective). Signup and view all the answers

CH 2 SUM: Functions

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Questions and Answers

What is the inverse function of $f(x) = 2x + 3$?

Which of the following is the general form of an exponential function?

What is the inverse function of $f(x) = 3^x$?

Which condition must be satisfied for a quadratic function to have an inverse?

What is the graphical representation of a linear function with a non-zero slope?

Which type of function is a one-to-one function?

What is the graphical representation of an exponential function with a base greater than 1?

Which test determines if a graph represents a function?

What is the condition for a function to have an inverse function?

Which of the following is an example of a many-to-one function?

What is the process for deriving the inverse of a linear function $y = ax + q$?

If $f(x) = -3x + 1$, what is the inverse function $f^{-1}(x)$?

What is the graphical relationship between a function and its inverse?

What happens to the domain and range when a linear function is inverted?

In the standard form of a quadratic function $y = ax^2 + bx + c$, what does the coefficient $a$ determine?

What is the axis of symmetry for a quadratic function?

What are the asymptotes of a hyperbolic function $y = a/(x - h) + k$?

What determines the orientation of a hyperbolic function?

What is the domain of a quadratic function?

How is the range of a hyperbolic function determined?

What is the defining characteristic of a function?

What is the defining characteristic of an inverse function?

What is the requirement for the existence of an inverse function?

What is the general form of a linear function?

Which of the following is NOT a fundamental concept in mathematical analysis?

What is the relationship between a function and its inverse function?

What is the inverse of a linear function expressed as $f(x) = 4x - 2$?

Which test is used to determine if a function has an inverse that is also a function?

For the function $f(x) = 5^x$, what is the inverse function?

What is the inverse of a quadratic function $f(x) = -3x^2$, considering $x$ non-negative?

In a linear function $f(x) = -2x + 5$, what does the value of '-2' represent?

What does the y-intercept signify in a linear function's graph?

For a linear function in the form $y = 3x - 7$, if you set $y$ to zero to find an intercept, which intercept are you calculating?

What is the domain and range of linear functions?

What does a positive slope 'a' indicate in a linear function?

How is the gradient (slope) calculated in a linear function?

What is the typical horizontal asymptote of an exponential function?

Which logarithmic law states that $\log_b(x^p) = p \cdot \log_b(x)$?

How does the base $b$ of an exponential function $f(x) = b^x$ affect the function's behavior?

What is the relationship between exponential and logarithmic functions?

Which transformation can be applied to both exponential and logarithmic graphs?

How can exponential equations be solved?

What is the domain of a logarithmic function $y = \log_b(x)$?

What is the y-intercept of an exponential function $f(x) = b^x$?

Which logarithmic law states that $\log_b(xy) = \log_b(x) + \log_b(y)$?

What is the typical vertical asymptote of a logarithmic function $y = \log_b(x)$?

What is the defining characteristic of an inverse function?

Which condition must be met for an inverse function to exist?

In a linear function $f(x) = 4x - 2$, what does the value '4' represent?

What is the role of a linear function's y-intercept?

Which type of function must a linear function be to have an inverse?

How is an inverse function symbolically denoted?

What is the key characteristic that determines the direction of a parabolic function?

Which of the following is NOT a characteristic of the inverse of a linear function?

What is the relationship between the vertex of a quadratic function and the axis of symmetry?

What is the defining characteristic of a hyperbolic function?

How does the domain and range of a quadratic function compare to the domain and range of its inverse?

What is the relationship between the $y$-intercept of a linear function and the $x$-intercept of its inverse?

What is the relationship between the $x$-intercept of a linear function and the $y$-intercept of its inverse?

What is the general form of a hyperbolic function?

Which of the following is NOT a characteristic of the axis of symmetry in a quadratic function?

What is the relationship between the domain and range of a hyperbolic function and its asymptotes?

What is the inverse of the quadratic function $f(x) = 4x^2$ when considering $x$ to be non-negative?

For a linear function in the form $y = -3x + 2$, what role does the value '-3' play?

In a linear function $y = 4x - 7$, setting $y$ to zero to find an intercept determines which intercept?

What does the Horizontal Line Test confirm regarding a function's graph?

Which characteristic does a negative slope in a linear function signify?

What is the inverse of an exponential function $f(x) = e^x$?

For a linear function $f(x) = mx + c$, what does the value of $m$ represent?

What is the inverse function of $f(x) = 2x^2 + 3$?

Which test is used to determine if a function is one-to-one (injective)?

For the exponential function $f(x) = 2^x$, what is the inverse function?

What is the condition for a function to have an inverse that is also a function?

What is the graphical representation of the inverse of a function?

For a quadratic function $f(x) = ax^2 + bx + c$, what does the coefficient $a$ determine?

Which type of function is a one-to-one function?