CH 2: Inverse functions

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What is the main purpose of inverse functions?

To reverse the action of original functions (correct)
To make functions more confusing
To complicate mathematical operations
To introduce errors in calculations

What property of a function is essential for the existence of an inverse function?

Multiplicity
Symmetry
Bijectivity (correct)
Surjectivity

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

They are parallel lines
They have the same shape
They are reflections across the line y = x (correct)
They intersect at every point

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

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

What happens when you compose a function with its inverse?

You get the identity function (C) Signup and view all the answers

In terms of mappings, what must be true for an inverse function to exist?

Each element in the range maps to exactly one element in the domain (C) Signup and view all the answers

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

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

Which of the following is the inverse function of the quadratic function $f(x) = ax^2$ with $a > 0$ and $x \geq 0$?

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

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

Horizontal Line Test (D) Signup and view all the answers

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

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

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

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

What is the inverse function of the quadratic function $f(x) = 2x^2$ with $x \geq 0$?

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

Which test is used to verify if a relation is a function?

Vertical Line Test (A) 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 injective (one-to-one) (C) Signup and view all the answers

Which of the following is true about the relationship between exponential and logarithmic functions?

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

How can a quadratic function $f(x) = ax^2$ have an inverse function?

By restricting the domain to $x \geq 0$ (D) Signup and view all the answers

What is the essential property required for the existence of an inverse function?

Bijectivity (D) Signup and view all the answers

If a function is not bijective, what can be said about the existence of its inverse?

It has no inverse (D) Signup and view all the answers

What happens to the roles of inputs and outputs in inverse functions?

They are interchanged (B) Signup and view all the answers

Which line serves as the line of reflection between a function and its inverse in terms of graphical representation?

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

In the graph of a function and its inverse, what is the relationship between their points?

They are symmetrical about the line $y = x$ (D) Signup and view all the answers

For a linear function $f(x) = mx + c$, what property should $m$ possess for the inverse to exist?

$m \neq 0$ (C) Signup and view all the answers

What is the result when you compose a function with its inverse?

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

In terms of bijective functions, what does 'onto' mean?

'Onto' means each element in the codomain is connected to at least one element in the domain (D) Signup and view all the answers

If a function has an inverse, what can be said about its graph with respect to the line $y=x$?

$y=x$ intersects the graph once (C) Signup and view all the answers

What function represents the inverse of a quadratic function $f(x) = ax^2$, where $a > 0$?

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

Which of the following describes the relationship between exponential and logarithmic functions?

They are inverses of each other. (A) Signup and view all the answers

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

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

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

Horizontal Line Test (A) Signup and view all the answers

For the function $f(x) = 4^x$, what is its inverse function?

$f^{-1}(x) = \log_4(x)$ (C) Signup and view all the answers

What condition must be satisfied by a function to ensure the existence of its inverse?

It must pass the Horizontal Line Test. (C) Signup and view all the answers

Which of the following represents the inverse of a linear function $f(x) = -5x + 2$?

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

What is the inverse of the function $f(x) = e^x$, where $e$ is Euler's number?

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

For a quadratic function $f(x) = 3x^2$, with $x eq 0$, what would be its inverse function?

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

What is a key aspect that must be present for the existence of an inverse function?

Injectivity (C) Signup and view all the answers

In terms of graph theory, what type of transformation occurs between the graphs of a function and its inverse?

Reflection over the line y = x (B) Signup and view all the answers

For a linear function f(x) = mx + c with m = 0, what property prevents the existence of its inverse?

Injectivity (C) Signup and view all the answers

What is an essential condition that guarantees the existence of an inverse function?

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

What feature in a function's graph signifies the existence of its inverse function?

Symmetry with respect to the y-axis (A) Signup and view all the answers

If a function is not one-to-one, what aspect prevents it from having an inverse function?

$f$ having more than one element in its domain mapping to the same element in its range (C) Signup and view all the answers

What is the consequence of a function's range having elements connecting to more than one element in its domain?

$f$ not having an inverse (D) Signup and view all the answers

Which feature distinguishes bijective functions regarding their inverses?

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

$f(x) = |x|$ is not invertible. Which property primarily prevents $f(x)$ from having an inverse?

$f$ having multiple elements mapping to the same element in its range (B) Signup and view all the answers

'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?

'One-to-one and onto' (C) Signup and view all the answers

Which statement is true about the inverse of a quadratic function $f(x) = ax^2$, where $a > 0$ and $x \geq 0?

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

For a function $f(x)$ to have an inverse that is also a function, which condition must be satisfied?

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

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

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

If a function $f(x)$ is not bijective (one-to-one and onto), what can be said about the existence of its inverse?

The inverse function exists but is not a function (A) Signup and view all the answers

What is the inverse function of the linear function $f(x) = 3x - 5$?

$f^{-1}(x) = \frac{x - 5}{3}$ (B) Signup and view all the answers

What is the result of composing a function $f(x)$ with its inverse $f^{-1}(x)$?

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

For a function $f(x)$ to have an inverse, what property must it possess?

It must be injective (one-to-one) (A) Signup and view all the answers

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

Horizontal Line Test (HLT) (C) Signup and view all the answers

What is the relationship between exponential and logarithmic functions?

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

What is the inverse function of the exponential function $f(x) = e^x$, where $e$ is Euler's number?

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

What is the key characteristic of bijectivity that is crucial for the existence of an inverse function?

Ensuring each element in the range is linked to only one element in the domain (C) Signup and view all the answers

If a function is not bijective, what aspect prevents it from having an inverse function?

Lacking a one-to-one correspondence between elements in the domain and range (A) Signup and view all the answers

What transformation occurs between the graphs of a function and its inverse?

Reflection across the line y = x (D) Signup and view all the answers

For a linear function f(x) = mx + c with m = 0, what property prevents the existence of its inverse?

Having a constant slope of 0 (A) Signup and view all the answers

What does 'onto' signify in the context of bijective functions?

Each element in the range is connected to only one element in the domain (C) Signup and view all the answers

What occurs to the roles of inputs and outputs in inverse functions?

Inputs and outputs switch places (C) Signup and view all the answers

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

$m eq 0$ (B) Signup and view all the answers

Which property distinguishes bijective functions regarding their inverses?

'One-to-one' and 'onto' mapping capabilities (B) Signup and view all the answers

Which test is used to determine if a relation is a function?

'Vertical line test' (C) Signup and view all the answers

'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?

'Bijection' (A) Signup and view all the answers

What is the inverse function of $f(x) = 4x^2$, considering $x eq 0$?

$f^{-1}(x) = rac{1}{4} imes ext{square root of } x$ (C) Signup and view all the answers

Which function represents the inverse of $f(x) = 5^x$?

$f^{-1}(x) = ext{log}_5(x)$ (A) Signup and view all the answers

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

$f^{-1}(x) = - ext{square root of } x/2$ (A), $f^{-1}(x) = - ext{square root of } x/2$ (C) Signup and view all the answers

What is the inverse function of $f(x) = e^{2x}$, where $e$ is Euler's number?

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

Given $f(x) = 6^x$, what is the correct representation for its inverse function?

$f^{-1}(x) = ext{log}_6(x)$ (C) Signup and view all the answers

What is the correct inverse function for $f(x) = -3x^2$, given that $x > 0$?

$f^{-1}(x) = - ext{square root of } x/3$ (A), $f^{-1}(x) = - ext{square root of } x/3$ (C) Signup and view all the answers

For $f(x) = 4^x$, what is the correct representation for its inverse function?

$f^{-1}(x) = ext{log}_4(x)$ (B) Signup and view all the answers

What represents the inverse function of $f(x) = 7^x$?

$f^{-1}(x) = ext{log}_7(7)$ (D) Signup and view all the answers

If $g(x) = -5^x$, what is the correct inverse function representation?

$g^{-1}(x) = - ext{log}_5(-x)$ (B) Signup and view all the answers

What would be the correct inverse function for $h(x) = 8^x$?

$h^{-1}(x) = ext{log}_8(8)$ (B) Signup and view all the answers

CH 2: Inverse functions

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Questions and Answers

What is the main purpose of inverse functions?

What property of a function is essential for the existence of an inverse function?

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

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

What happens when you compose a function with its inverse?

In terms of mappings, what must be true for an inverse function to exist?

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

Which of the following is the inverse function of the quadratic function $f(x) = ax^2$ with $a > 0$ and $x \geq 0$?

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

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

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

What is the inverse function of the quadratic function $f(x) = 2x^2$ with $x \geq 0$?

Which test is used to verify if a relation is a function?

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

Which of the following is true about the relationship between exponential and logarithmic functions?

How can a quadratic function $f(x) = ax^2$ have an inverse function?

What is the essential property required for the existence of an inverse function?

If a function is not bijective, what can be said about the existence of its inverse?

What happens to the roles of inputs and outputs in inverse functions?

Which line serves as the line of reflection between a function and its inverse in terms of graphical representation?

In the graph of a function and its inverse, what is the relationship between their points?

For a linear function $f(x) = mx + c$, what property should $m$ possess for the inverse to exist?

What is the result when you compose a function with its inverse?

In terms of bijective functions, what does 'onto' mean?

If a function has an inverse, what can be said about its graph with respect to the line $y=x$?

What function represents the inverse of a quadratic function $f(x) = ax^2$, where $a > 0$?

Which of the following describes the relationship between exponential and logarithmic functions?

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

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

For the function $f(x) = 4^x$, what is its inverse function?

What condition must be satisfied by a function to ensure the existence of its inverse?

Which of the following represents the inverse of a linear function $f(x) = -5x + 2$?

What is the inverse of the function $f(x) = e^x$, where $e$ is Euler's number?

For a quadratic function $f(x) = 3x^2$, with $x eq 0$, what would be its inverse function?

What is a key aspect that must be present for the existence of an inverse function?

In terms of graph theory, what type of transformation occurs between the graphs of a function and its inverse?

For a linear function f(x) = mx + c with m = 0, what property prevents the existence of its inverse?

What is an essential condition that guarantees the existence of an inverse function?

What feature in a function's graph signifies the existence of its inverse function?

If a function is not one-to-one, what aspect prevents it from having an inverse function?

What is the consequence of a function's range having elements connecting to more than one element in its domain?

Which feature distinguishes bijective functions regarding their inverses?

$f(x) = |x|$ is not invertible. Which property primarily prevents $f(x)$ from having an inverse?

'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?

Which statement is true about the inverse of a quadratic function $f(x) = ax^2$, where $a > 0$ and $x \geq 0?

For a function $f(x)$ to have an inverse that is also a function, which condition must be satisfied?

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

If a function $f(x)$ is not bijective (one-to-one and onto), what can be said about the existence of its inverse?

What is the inverse function of the linear function $f(x) = 3x - 5$?

What is the result of composing a function $f(x)$ with its inverse $f^{-1}(x)$?

For a function $f(x)$ to have an inverse, what property must it possess?

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

What is the relationship between exponential and logarithmic functions?

What is the inverse function of the exponential function $f(x) = e^x$, where $e$ is Euler's number?

What is the key characteristic of bijectivity that is crucial for the existence of an inverse function?

If a function is not bijective, what aspect prevents it from having an inverse function?

What transformation occurs between the graphs of a function and its inverse?

For a linear function f(x) = mx + c with m = 0, what property prevents the existence of its inverse?

What does 'onto' signify in the context of bijective functions?

What occurs to the roles of inputs and outputs in inverse functions?

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

Which property distinguishes bijective functions regarding their inverses?

Which test is used to determine if a relation is a function?

'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?

What is the inverse function of $f(x) = 4x^2$, considering $x eq 0$?

Which function represents the inverse of $f(x) = 5^x$?

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

What is the inverse function of $f(x) = e^{2x}$, where $e$ is Euler's number?

Given $f(x) = 6^x$, what is the correct representation for its inverse function?

What is the correct inverse function for $f(x) = -3x^2$, given that $x > 0$?

For $f(x) = 4^x$, what is the correct representation for its inverse function?

What represents the inverse function of $f(x) = 7^x$?

If $g(x) = -5^x$, what is the correct inverse function representation?

What would be the correct inverse function for $h(x) = 8^x$?

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