CH 2: Linear functions

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What does the parameter 'a' represent in a linear function of the form $y = ax + q$?

The domain of the function

The range of the function

The y-intercept

The gradient (slope) of the line (correct)

For a linear function, what is the relationship between the domain and range?

The domain is an infinite set, while the range is a finite set

The domain is a finite set, while the range is an infinite set

The domain and range are both finite sets

The domain and range are both infinite sets (correct)

What is the purpose of the y-intercept 'q' in a linear function of the form $y = ax + q$?

It represents the domain of the function

It specifies the point where the line intersects the y-axis (correct)

It determines the slope of the line

It specifies the point where the line intersects the x-axis

Which method can be used to construct the graph of a linear function?

Identifying the intercepts of the line Signup and view all the answers

What is the first step in the process of deriving the inverse of a linear function?

Interchange x and y Signup and view all the answers

What is the key difference between a linear function and its inverse?

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

How does the slope 'a' of a linear function affect the direction of the line?

Positive values of 'a' lead to an upward trajectory, while negative values result in a downward trend Signup and view all the answers

What is the purpose of the 'gradient-intercept approach' in constructing the graph of a linear function?

To locate a secondary point on the line using the y-intercept and slope Signup and view all the answers

Which of the following is NOT a key mathematical attribute of a linear function?

Derivative Signup and view all the answers

What is the purpose of deriving the inverse of a linear function?

To invert the function's operation, swapping the roles of inputs and outputs Signup and view all the answers

What is the inverse function of $f(x) = -3x + 1$?

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

How are the graphs of a function and its inverse related?

They are mirrored along the line $y = x$ Signup and view all the answers

What happens to the domain and range of a function when finding its inverse?

The domain becomes the range, and the range becomes the domain Signup and view all the answers

In the example $f(x) = 2x - 3$, what is the y-intercept of the inverse function $f^{-1}(x)$?

(-3, 0) Signup and view all the answers

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

The function must be bijective Signup and view all the answers

What is the significance of the line $y = x$ in the context of functions and their inverses?

It is the symmetry axis along which the functions are mirrored Signup and view all the answers

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

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

In the example $f(x) = 2x - 3$, what is the domain and range of the inverse function $f^{-1}(x)$?

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

What is the x-intercept of the inverse function $f^{-1}(x) = (x/2) + 1.5$ in the example given?

(0, 1.5) Signup and view all the answers

What is the significance of studying linear functions and their inverses?

It helps in understanding constant relationships Signup and view all the answers

What is the general formula for the inverse of a linear function $f(x) = ax + q$?

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

In the context of linear functions and their inverses, what transformation occurs between the original and the inverse with respect to intercepts?

X and Y intercepts switch positions Signup and view all the answers

How do the graphs of a linear function and its inverse relate to each other geometrically?

They mirror each other across the line $y = x$ Signup and view all the answers

When considering the linearity of functions and their inverses, what condition must be satisfied for both to be true linear functions?

Being bijective Signup and view all the answers

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

It is the symmetry axis across which they reflect each other Signup and view all the answers

In a linear function $f(x) = 2x - 3$, what are the domain and range values?

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

What is the significant change observed in the intercepts when transitioning from a linear function to its inverse?

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

Which statement accurately describes the uniqueness of linear functions and their inverses?

$f(x)$ and $f^{-1}(x)$ are both linear under bijective conditions Signup and view all the answers

When considering linear functions and their inverses, how does the slope 'a' impact their behavior with respect to linearity?

$a$ determines the steepness or inclination of the lines Signup and view all the answers

In general, what happens to a function's domain and range when finding its inverse?

$\text{Domain} \longleftrightarrow \text{Range}$ interchange positions Signup and view all the answers

What is the general form of a linear function?

y = ax + q Signup and view all the answers

Which of the following statements about the domain and range of linear functions is correct?

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

What does the slope 'a' in the linear function y = ax + q represent?

The angle and direction of the line's tilt Signup and view all the answers

Which method can be used to construct the graph of a linear function by identifying two points?

Gradient-Intercept Approach Signup and view all the answers

If f(x) = 3x - 2, what is the inverse function f^(-1)(x)?

(x - 2)/3 Signup and view all the answers

What is the purpose of studying the inverses of linear functions?

To invert the operation of the function, swapping inputs and outputs Signup and view all the answers

If the slope of a linear function is zero, what can be inferred about the function?

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

Which of the following statements about the graph of a linear function and its inverse is true?

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

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

The slope 'a' must be non-zero Signup and view all the answers

What is the significance of the line y = x in the context of functions and their inverses?

It represents the line of reflection for a function and its inverse Signup and view all the answers

What characteristic defines the relationship between a linear function and its inverse when graphed?

They reflect each other across the line y = x. Signup and view all the answers

In the context of linear functions and their inverses, what happens to the x and y intercepts when transitioning from a linear function to its inverse?

They switch positions. Signup and view all the answers

What is the symbolic significance of the line y = x in relation to linear functions and their inverses?

It acts as a mirroring axis for functions and their inverses. Signup and view all the answers

Which aspect ensures that both a linear function and its inverse maintain linearity?

Bijective property. Signup and view all the answers

What key transformation occurs regarding domain and range when transitioning from a linear function to its inverse?

Both domain and range remain unchanged. Signup and view all the answers

In terms of intercepts, what change is observed when moving from a linear function to its inverse?

Intercepts switch positions. Signup and view all the answers

What geometric relationship is symbolized by a linear function and its inverse across the line y = x?

Reflection across a line Signup and view all the answers

Why is it important for linear functions and their inverses to maintain linearity?

To retain fundamental properties of linear functions. Signup and view all the answers

What characterizes the symmetry between a linear function and its inverse graphically?

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

If $f(x) = 2x - 5$ and $g(x) = x/2 + 3$, which of the following statements is true?

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

If $f(x) = ax + b$ is a linear function, and $f^{-1}(x)$ is its inverse, what is the value of $a$ if $f(f^{-1}(x)) = x$ for all $x$ in the domain?

$a = 1$ Signup and view all the answers

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

The slopes are reciprocals of each other. Signup and view all the answers

If the graph of a linear function $f(x)$ passes through the points $(2, 5)$ and $(4, 9)$, what is the equation of its inverse function $f^{-1}(x)$?

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

If $f(x) = 3x - 2$ and $g(x) = (x + 2)/3$, which of the following statements is true?

Both $f(g(x)) = x$ and $g(f(x)) = x$ are true. Signup and view all the answers

If the graph of a linear function $f(x)$ passes through the points $(0, -3)$ and $(2, 1)$, what is the equation of its inverse function $f^{-1}(x)$?

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

If $f(x) = ax + b$ is a linear function and $f^{-1}(x)$ is its inverse, what is the relationship between the y-intercepts of $f(x)$ and $f^{-1}(x)$?

The y-intercepts are additive inverses. Signup and view all the answers

If $f(x) = 2x - 1$ and $g(x) = x/2 + 1$, which of the following statements is true?

Both $f(g(x)) = x$ and $g(f(x)) = x$ are true. Signup and view all the answers

If the graph of a linear function $f(x)$ passes through the points $(1, 4)$ and $(3, 8)$, what is the equation of its inverse function $f^{-1}(x)$?

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

If $f(x) = ax + b$ is a linear function and $f^{-1}(x)$ is its inverse, what is the relationship between the slopes of $f(x)$ and $f^{-1}(x)$ if $f(x)$ is a horizontal line?

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

CH 2: Linear functions

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Questions and Answers

What does the parameter 'a' represent in a linear function of the form $y = ax + q$?

For a linear function, what is the relationship between the domain and range?

What is the purpose of the y-intercept 'q' in a linear function of the form $y = ax + q$?

Which method can be used to construct the graph of a linear function?

What is the first step in the process of deriving the inverse of a linear function?

What is the key difference between a linear function and its inverse?

How does the slope 'a' of a linear function affect the direction of the line?

What is the purpose of the 'gradient-intercept approach' in constructing the graph of a linear function?

Which of the following is NOT a key mathematical attribute of a linear function?

What is the purpose of deriving the inverse of a linear function?

What is the inverse function of $f(x) = -3x + 1$?

How are the graphs of a function and its inverse related?

What happens to the domain and range of a function when finding its inverse?

In the example $f(x) = 2x - 3$, what is the y-intercept of the inverse function $f^{-1}(x)$?

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

What is the significance of the line $y = x$ in the context of functions and their inverses?

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

In the example $f(x) = 2x - 3$, what is the domain and range of the inverse function $f^{-1}(x)$?

What is the x-intercept of the inverse function $f^{-1}(x) = (x/2) + 1.5$ in the example given?

What is the significance of studying linear functions and their inverses?

What is the general formula for the inverse of a linear function $f(x) = ax + q$?

In the context of linear functions and their inverses, what transformation occurs between the original and the inverse with respect to intercepts?

How do the graphs of a linear function and its inverse relate to each other geometrically?

When considering the linearity of functions and their inverses, what condition must be satisfied for both to be true linear functions?

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

In a linear function $f(x) = 2x - 3$, what are the domain and range values?

What is the significant change observed in the intercepts when transitioning from a linear function to its inverse?

Which statement accurately describes the uniqueness of linear functions and their inverses?

When considering linear functions and their inverses, how does the slope 'a' impact their behavior with respect to linearity?

In general, what happens to a function's domain and range when finding its inverse?

What is the general form of a linear function?

Which of the following statements about the domain and range of linear functions is correct?

What does the slope 'a' in the linear function y = ax + q represent?

Which method can be used to construct the graph of a linear function by identifying two points?

If f(x) = 3x - 2, what is the inverse function f^(-1)(x)?

What is the purpose of studying the inverses of linear functions?

If the slope of a linear function is zero, what can be inferred about the function?

Which of the following statements about the graph of a linear function and its inverse is true?

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

What is the significance of the line y = x in the context of functions and their inverses?

What characteristic defines the relationship between a linear function and its inverse when graphed?

In the context of linear functions and their inverses, what happens to the x and y intercepts when transitioning from a linear function to its inverse?

What is the symbolic significance of the line y = x in relation to linear functions and their inverses?

Which aspect ensures that both a linear function and its inverse maintain linearity?

What key transformation occurs regarding domain and range when transitioning from a linear function to its inverse?

In terms of intercepts, what change is observed when moving from a linear function to its inverse?

What geometric relationship is symbolized by a linear function and its inverse across the line y = x?

Why is it important for linear functions and their inverses to maintain linearity?

What characterizes the symmetry between a linear function and its inverse graphically?

If $f(x) = 2x - 5$ and $g(x) = x/2 + 3$, which of the following statements is true?

If $f(x) = ax + b$ is a linear function, and $f^{-1}(x)$ is its inverse, what is the value of $a$ if $f(f^{-1}(x)) = x$ for all $x$ in the domain?

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

If the graph of a linear function $f(x)$ passes through the points $(2, 5)$ and $(4, 9)$, what is the equation of its inverse function $f^{-1}(x)$?

If $f(x) = 3x - 2$ and $g(x) = (x + 2)/3$, which of the following statements is true?

If the graph of a linear function $f(x)$ passes through the points $(0, -3)$ and $(2, 1)$, what is the equation of its inverse function $f^{-1}(x)$?

If $f(x) = ax + b$ is a linear function and $f^{-1}(x)$ is its inverse, what is the relationship between the y-intercepts of $f(x)$ and $f^{-1}(x)$?

If $f(x) = 2x - 1$ and $g(x) = x/2 + 1$, which of the following statements is true?

If the graph of a linear function $f(x)$ passes through the points $(1, 4)$ and $(3, 8)$, what is the equation of its inverse function $f^{-1}(x)$?

If $f(x) = ax + b$ is a linear function and $f^{-1}(x)$ is its inverse, what is the relationship between the slopes of $f(x)$ and $f^{-1}(x)$ if $f(x)$ is a horizontal line?

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