74 Questions
What is the main purpose of inverse functions?
To reverse the action of original functions
What property of a function is essential for the existence of an inverse function?
Bijectivity
How are the graphs of a function and its inverse related?
They are reflections across the line y = x
For a linear function f(x) = mx + c, what is the inverse function?
$f^{-1}(x) = \frac{x - c}{m}$
What happens when you compose a function with its inverse?
You get the identity function
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
What is the inverse function of the exponential function $f(x) = a^x$?
$f^{-1}(x) = \$\log_{a}(x)$
Which of the following is the inverse function of the quadratic function $f(x) = ax^2$ with $a > 0$ and $x \geq 0$?
$f^{-1}(x) = \$\sqrt{x/a}$
Which test is used to determine if a function is injective (one-to-one)?
Horizontal Line Test
What is the inverse function of the linear function $f(x) = 2x + 3$?
$f^{-1}(x) = (x - 3)/2$
What is the inverse function of the exponential function $f(x) = 3^x$?
$f^{-1}(x) = \$\log_3(x)$
What is the inverse function of the quadratic function $f(x) = 2x^2$ with $x \geq 0$?
$f^{-1}(x) = \$\sqrt{x/2}$
Which test is used to verify if a relation is a function?
Vertical Line Test
What is the condition for a function to have an inverse that is also a function?
The function must be injective (one-to-one)
Which of the following is true about the relationship between exponential and logarithmic functions?
Logarithmic functions are the inverse of exponential functions
How can a quadratic function $f(x) = ax^2$ have an inverse function?
By restricting the domain to $x \geq 0$
What is the essential property required for the existence of an inverse function?
Bijectivity
If a function is not bijective, what can be said about the existence of its inverse?
It has no inverse
What happens to the roles of inputs and outputs in inverse functions?
They are interchanged
Which line serves as the line of reflection between a function and its inverse in terms of graphical representation?
$y = x$
In the graph of a function and its inverse, what is the relationship between their points?
They are symmetrical about the line $y = x$
For a linear function $f(x) = mx + c$, what property should $m$ possess for the inverse to exist?
$m \neq 0$
What is the result when you compose a function with its inverse?
$f(f^{-1}(x)) = f^{-1}(f(x)) = x$
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
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
What function represents the inverse of a quadratic function $f(x) = ax^2$, where $a > 0$?
$f^{-1}(x) = \sqrt{\frac{x}{a}}$
Which of the following describes the relationship between exponential and logarithmic functions?
They are inverses of each other.
What is the inverse function of $f(x) = 2x + 4$?
$f^{-1}(x) = \frac{x + 4}{2}$
Which test is used to determine if a function is injective (one-to-one)?
Horizontal Line Test
For the function $f(x) = 4^x$, what is its inverse function?
$f^{-1}(x) = \log_4(x)$
What condition must be satisfied by a function to ensure the existence of its inverse?
It must pass the Horizontal Line Test.
Which of the following represents the inverse of a linear function $f(x) = -5x + 2$?
$f^{-1}(x) = \frac{5x - 2}{5}$
What is the inverse of the function $f(x) = e^x$, where $e$ is Euler's number?
$f^{-1}(x) = \ln(x)$
For a quadratic function $f(x) = 3x^2$, with $x eq 0$, what would be its inverse function?
$f^{-1}(x) = \sqrt{\frac{x}{3}}$
What is a key aspect that must be present for the existence of an inverse function?
Injectivity
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
For a linear function f(x) = mx + c with m = 0, what property prevents the existence of its inverse?
Injectivity
What is an essential condition that guarantees the existence of an inverse function?
$f^{-1}(f(x)) = x$
What feature in a function's graph signifies the existence of its inverse function?
Symmetry with respect to the y-axis
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
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
Which feature distinguishes bijective functions regarding their inverses?
$f(f^{-1}(y)) = y$ for all y
$f(x) = |x|$ 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
'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?
'One-to-one and onto'
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 $f^{-1}(x) = \sqrt{x/a}$
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)
What is the inverse function of the exponential function $f(x) = 2^x$?
$f^{-1}(x) = \log_2(x)$
If a function $f(x)$ is not bijective (one-to-one and onto), what can be said about the existence of its inverse?
The inverse function exists but is not a function
What is the inverse function of the linear function $f(x) = 3x - 5$?
$f^{-1}(x) = \frac{x - 5}{3}$
What is the result of composing a function $f(x)$ with its inverse $f^{-1}(x)$?
The identity function $f(f^{-1}(x)) = x$
For a function $f(x)$ to have an inverse, what property must it possess?
It must be injective (one-to-one)
Which test is used to determine if a function is injective (one-to-one)?
Horizontal Line Test (HLT)
What is the relationship between exponential and logarithmic functions?
They are inverses of each other
What is the inverse function of the exponential function $f(x) = e^x$, where $e$ is Euler's number?
$f^{-1}(x) = \log_e(x)$
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
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
What transformation occurs between the graphs of a function and its inverse?
Reflection across the line y = x
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
What does 'onto' signify in the context of bijective functions?
Each element in the range is connected to only one element in the domain
What occurs to the roles of inputs and outputs in inverse functions?
Inputs and outputs switch places
What is a necessary condition for a linear function to have an inverse?
$m eq 0$
Which property distinguishes bijective functions regarding their inverses?
'One-to-one' and 'onto' mapping capabilities
Which test is used to determine if a relation is a function?
'Vertical line test'
'Bijectivity' is crucial for an inverse function. Which term represents this bijective quality?
'Bijection'
What is the inverse function of $f(x) = 4x^2$, considering $x eq 0$?
$f^{-1}(x) = rac{1}{4} imes ext{square root of } x$
Which function represents the inverse of $f(x) = 5^x$?
$f^{-1}(x) = ext{log}_5(x)$
For $f(x) = -2x^2$, what is the correct inverse function?
$f^{-1}(x) = - ext{square root of } x/2$
What is the inverse function of $f(x) = e^{2x}$, where $e$ is Euler's number?
$f^{-1}(x) = ext{log}_e(2x)$
Given $f(x) = 6^x$, what is the correct representation for its inverse function?
$f^{-1}(x) = ext{log}_6(x)$
What is the correct inverse function for $f(x) = -3x^2$, given that $x > 0$?
$f^{-1}(x) = - ext{square root of } x/3$
For $f(x) = 4^x$, what is the correct representation for its inverse function?
$f^{-1}(x) = ext{log}_4(x)$
What represents the inverse function of $f(x) = 7^x$?
$f^{-1}(x) = ext{log}_7(7)$
If $g(x) = -5^x$, what is the correct inverse function representation?
$g^{-1}(x) = - ext{log}_5(-x)$
What would be the correct inverse function for $h(x) = 8^x$?
$h^{-1}(x) = ext{log}_8(8)$
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