Chapter 1.5, Problem 14E

To determine

To find: The inverse of the given function and also verify that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Answer to Problem 14E

The inverse of the given function is $f^{-1}\left(x\right)=\frac{5-x}{4}$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=5-4x$ .

Calculation:

Substitute $y$ for $f\left(x\right)$ in the function and solve the equation for $x$ in terms of $y$ .

  $\begin{array}{c}y=5-4x\\ 4x=5-y\\ x=\frac{5-y}{4}\end{array}$

Interchange $x$ and $y$ , and substitute $f^{-1}\left(x\right)$ for $y$ .

  $\begin{array}{c}y=\frac{5-x}{4}\\ f^{-1}\left(x\right)=\frac{5-x}{4}\end{array}$

Verify:

The composite function $\left(f\circ f^{-1}\right)\left(x\right)$ can be written as:

  $\left(f\circ f^{-1}\right)\left(x\right)=f\left(f^{-1}\left(x\right)\right)$

Substitute $\frac{5-x}{4}$ for $f^{-1}\left(x\right)$ to find $\left(f\circ f^{-1}\right)\left(x\right)$ .

  $\begin{array}{c}\left(f\circ f^{-1}\right)\left(x\right)=f\left(\frac{5-x}{4}\right)\\ =5-4\left(\frac{5-x}{4}\right)\\ =5-5+x\\ =x\end{array}$

The composite function $\left(f^{-1}\circ f\right)\left(x\right)$ can be written as:

  $\left(f^{-1}\circ f\right)\left(x\right)=f^{-1}\left(f\left(x\right)\right)$

Substitute $5-4x$ for $f\left(x\right)$ in to find $\left(f^{-1}\circ f\right)\left(x\right)$ .

  $\begin{array}{c}\left(f^{-1}\circ f\right)\left(x\right)=f^{-1}\left(5-4x\right)\\ =\frac{5-\left(5-4x\right)}{4}\\ =\frac{4x}{4}\\ =x\end{array}$

It is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Therefore, the inverse of the given function is $f^{-1}\left(x\right)=\frac{5-x}{4}$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .