Chapter 1.4, Problem 24E

Solution

To find: The functions $f\left(x\right)$ and $g\left(x\right)$ from the composition function $y=f\left(g(x)\right)$ .

The required functions are $f\left(x\right)=x^2$ and $g\left(x\right)=x^3+1$ .

Given data:

The function $y=f\left(g(x)\right)$ is $y={\left(x^3+1\right)}^2$ .

Method/Formula used:

The composition function of two functions $f\left(x\right)$ and $g\left(x\right)$ is denoted by $\left(f\text{ο}g\right)\left(x\right)$ is function defined by $\left(f\text{ο}g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Calculations:

The composite function $y={\left(x^3+1\right)}^2$ or $f\left(g\left(x\right)\right)={\left(x^3+1\right)}^2$ .

It is observed that $f\left(g\left(x\right)\right)$ is a square of the function $\left(x^3+1\right)$ , therefore, set $f\left(x\right)=x^2$ and let $g\left(x\right)=x^3+1$ . Then

  $\begin{array}{c}y=f\left(g\left(x\right)\right)\text{use}\text{definition}\text{of}g\left(x\right)\\ =f\left(x^3+1\right)\text{use}\text{definition}\text{of}f\left(x\right)\\ ={\left(x^3+1\right)}^2\end{array}$

Therefore, $f\left(x\right)=x^2$ and $g\left(x\right)=x^3+1$ .