Chapter 1.6, Problem 121E

To determine

To find : the value of function.

Answer to Problem 121E

The value of function ${\left(fog\right)}^{-1}\left(x\right)$ is $\frac{x+1}{2}$.

Explanation of Solution

Given information :

The functions are $f\left(x\right)=x+4$ and $g\left(x\right)=2x-5$.

The function for finding value ${\left(fog\right)}^{-1}\left(x\right)$.

Calculation : to find the value of function, it is needed to first find the inverse functions of $f\left(x\right)\text{ and }g\left(x\right)$.

To find the inverse function of $f\left(x\right)$ , substitute $f\left(x\right)\text{=}y$ in the function,

  $\begin{array}{l}f\left(x\right)=x+4\\ y=x+4\text{ [substitute }f\left(x\right)=y]\\ x=y-4\text{ [subtract 4 from both side]}\\ f^{-1}\left(x\right)=x-4\text{ [simplify]}\end{array}$

To find the inverse function of $g\left(x\right)$ , substitute $g\left(x\right)\text{=}y$ in the function,

  $\begin{array}{l}g\left(x\right)=2x-5\\ y=2x-5\text{ [substitute }g\left(x\right)=y]\\ 2x=y+5\text{ [add 5 both side]}\\ x=\frac{y+5}{2}\text{ [divide by 2]}\\ g^{-1}\left(x\right)=\frac{x+5}{2}\text{ [simplify]}\end{array}$

Since it is known that ${\left(fog\right)}^{-1}\left(x\right)=\left(g^{-1}o{f}^{-1}\right)\left(x\right)$ ,

So find the value of ${\left(fog\right)}^{-1}\left(x\right)$ first, find the value of function $\left(g^{-1}o{f}^{-1}\right)\left(x\right)$ ,

  $\begin{array}{l}\left(g^{-1}o{f}^{-1}\right)\left(x\right)=g^{-1}\left(f^{-1}\left(x\right)\right)\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=g^{-1}\left(x-4\right)\text{ [ }f\left(x\right)=x-4]\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=\frac{x-4+5}{2}\text{ [substitute }x=x-4\text{ in }{g}^{-1}\left(x\right)=\frac{x+5}{2}]\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=\frac{x+1}{2}\text{ [simplify]}\end{array}$

Thus, The value of function ${\left(fog\right)}^{-1}\left(x\right)$ is $\frac{x+1}{2}$.