Chapter 1.4, Problem 29E

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 $f\left(x\right)=\cos \left(x\right)$ and $g\left(x\right)=\sqrt{x}$ .

Given data:

The function $y=f\left(g(x)\right)$ is $y=\cos \left(\sqrt{x}\right)$ .

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 also a function defined by $\left(f\text{ο}g\right)\left(x\right)=f\left(g\left(x\right)\right)$ .

Calculations:

The composite function $y=\cos \left(\sqrt{x}\right)$ or $f\left(g\left(x\right)\right)=\cos \left(\sqrt{x}\right)$ .

It is observed that $y=f\left(g\left(x\right)\right)$ is a cosine function of the form $\cos \left(x\right)$ therefore, set $f\left(x\right)=\cos \left(x\right)$ and let $g\left(x\right)=\sqrt{x}$ . Then

  $\begin{array}{c}y=f\left(g\left(x\right)\right)\leftarrow \text{use}\text{definition}\text{of}g\left(x\right)\\ =f\left(\sqrt{x}\right)\leftarrow \text{use}\text{definition}\text{of}f\left(x\right)\\ =\cos \left(\sqrt{x}\right)\end{array}$

Therefore, $f\left(x\right)=\cos \left(x\right)$ and $g\left(x\right)=\sqrt{x}$ .