AP®︎ Precalculus: Topic 2.8 Practice Test

Essential Knowledge: 2.8.A.1

On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible.

Essential Knowledge: 2.8.A.2

An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a)=b$, then $f^{-1}(b)=a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a,b)$, then the inverse function consists of input-output pairs $(b,a)$.

Essential Knowledge: 2.8.B.1

The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.

Essential Knowledge: 2.8.B.2

On a function’s invertible domain, the function’s range and domain are the inverse function’s domain and range, respectively. The inverse of the table of values of $y=f(x)$ can be found by reversing the input-output pairs; that is, $(a,b)$ corresponds to $(b,a)$.

Essential Knowledge: 2.8.B.3

The inverse of the graph of the function $y=f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x)=x$.

Essential Knowledge: 2.8.B.4

The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y=f(x)$, then solving for $y=f^{-1}(x)$.

Essential Knowledge: 2.8.B.5

In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.