Chapter 1.6, Problem 16E

To determine

To find: The value of $f^{-1}\left(3\right)\text{ and }f\left(f^{-1}\left(2\right)\right)$ if $f\left(x\right)=x^5+x^3+x$.

Answer to Problem 16E

The value of $f^{-1}\left(3\right)=1$.

The value of $f\left(f^{-1}\left(2\right)\right)=2$.

Explanation of Solution

The given function is $f\left(x\right)=x^5+x^3+x$.

Notice that the value of $f\left(1\right)=3$ as, $1^5+1^3+1=3$.

According to the definition of an inverse function, $f^{-1}\left(y\right)=x$ if and only if $f\left(x\right)=y$ where f is one-to-one.

The value of $f^{-1}\left(3\right)=1$ since $f\left(1\right)=3$.

Express $f\left(f^{-1}\left(2\right)\right)$ as $f^{-1}\left(f\left(2\right)\right)$.

Apply the cancellation equation of the inverse function and obtain $f^{-1}\left(f\left(2\right)\right)=2$.