True or False? An inverse function reverses the effect of a function .

True. An inverse function reverses the effect of a function .

True or False? Only a many-to-one function has an inverse function.

False. A many-to-one function does not have an inverse function. Only one-to-one functions have inverses .

What is the horizontal line test ?

The horizontal line test is a method to determine if a function has an inverse . If a function has an inverse , then any horizontal line should intersect with the graph once at most .

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

The domain of a function becomes the range of its inverse function .

True or False? For a many-to-one function with a restricted domain , the range of its inverse function is the same as the original function 's domain .

False. For a many-to-one function with a restricted domain, the range of its inverse function is the not the same as the original function's domain. The range of the inverse function is the same as the restricted domain of the original function .

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FrontComposite & Inverse Functions
What is a composite function?

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What is a composite function?
A composite function is a function made up of two or more functions. One function is applied to another function, e.g. $(f \circ g) (x)$ .
True or False?
The order in which the functions in a composite function are applied does not matter.
False.
The order in which the functions in a composite function are applied does matter. $(f \circ g) (x)$ is not usually equivalent to $(g \circ f) (x)$ .
Always apply the function closest to the variable first.
E.g. for the composite function $(f \circ g) (x)$ , you must first apply the function $g$ followed by the function $f$ .
To find the domain and range of a composite function, what is the first step you need to take?
To find the domain and range of a composite function, the first step is you need to take is to find the range of the first function that is applied.
E.g. To find the domain and range for the function $(f \circ g) (x)$ , you must first find the range of $g$ .
True or False?
An inverse function reverses the effect of a function.
True.
An inverse function reverses the effect of a function.
True or False?
Only a many-to-one function has an inverse function.
False.
A many-to-one function does not have an inverse function.
Only one-to-one functions have inverses.
What is the horizontal line test?
The horizontal line test is a method to determine if a function has an inverse.
If a function has an inverse, then any horizontal line should intersect with the graph once at most.
What is the relationship between the domain of a function and its inverse function?
The domain of a function becomes the range of its inverse function.
State the relationship between the graphs of $f (x)$ and $f^{- 1} (x)$ .
The graph $y = f^{- 1} (x)$ is a reflection of the graph $y = f (x)$ in the line $y = x$ .
What is the identity function?
The identity function maps each value in a function to itself.
$id (x) = x$ .
If two composite functions, $f \circ g$ and $g \circ f$ , have the same effect as the identity function, what can be said about the functions $f$ and $g$ ?
If two composite functions, $f \circ g$ and $g \circ f$ , have the same effect as the identity function, then the functions $f$ and $g$ must be inverse functions of each other.
How can you adapt a many-to-one function in order for it to have an inverse?
You can restrict the domain of a many-to-one function to a subset of the domain on which the function is one-to-one. Then the function on the restricted domain will have an inverse.
E.g. the function $y = x^{2}$ is a many-to-one function, but if the domain is restricted to $x \geq 0$ , the function becomes one-to-one and has an inverse.
How should the domain for the sine function be restricted in order for it to have an inverse?
In order for the sine function to have an inverse, the domain must be restricted to half a cycle, from a maximum to a minimum or vice versa.
E.g. $- \frac{π}{2} \leq x \leq \frac{π}{2}$ .
True or False?
For a many-to-one function with a restricted domain, the range of its inverse function is the same as the original function's domain.
False.
For a many-to-one function with a restricted domain, the range of its inverse function is the not the same as the original function's domain.
The range of the inverse function is the same as the restricted domain of the original function.

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