Chapter 1.9, Problem 97E

To determine

Find value of k in the given function.

Answer to Problem 97E

The value of k in the given function $k=\frac{1}{4}$

Explanation of Solution

Given:

The following

  $f\left(x\right)=k\left(2–x–x^3\right)$ and

  $f^{-1}\left(3\right)=–2$

Let consider the following

  $f\left(x\right)=k\left(2–x–x^3\right)$ and

  $f^{-1}\left(3\right)=–2$

Since the function $f\left(x\right)$ has an inverse function

  $f\left(–2\right)=3$ or

Put value of x, that is −2, in the given function

  $\begin{array}{l}k\left(2–\left(–2\right)–\left(–2\right)3\right))=3\hfill \\ k\left(2+2+8\right)=3\hfill \\ 12k=3\hfill \\ k=\frac{1}{4}\hfill \end{array}$