Chapter 1.4, Problem 13E

Solution

To find: The functions $fog\left(x\right)$ and $gof\left(x\right)$ from the functions $f\left(x\right)$ and $g\left(x\right)$ .

The required functions values are $fog\left(3\right)=8$ and $gof\left(-2\right)=3$ .

Given data:

The functions $f\left(x\right)$ and $g\left(x\right)$ are; $f\left(x\right)=x^2+4$ and $g\left(x\right)=\sqrt{x+1}$ . The required functions are $fog\left(3\right)$ and $gof\left(-2\right)$ .

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 $\left(f\text{ο}g\right)\left(x\right)=f\left(g\left(x\right)\right)$ . Similarly, $\left(g\text{ο}f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Calculations:

The given functions $f\left(x\right)$ and $g\left(x\right)$ are $f\left(x\right)=x^2+4$ and $g\left(x\right)=\sqrt{x+1}$ . Then by the definition of $fog\left(x\right)$ ,

  $\begin{array}{c}fog\left(x\right)=f\left(g\left(x\right)\right)\leftarrow \text{use}\text{definition}\text{of}g\left(x\right)\text{, i}\text{.e}\text{., rereplace}g\left(x\right)\text{by}\sqrt{x+1}\\ =f\left(\sqrt{x+1}\right)\leftarrow \text{use}\text{definition}\text{of}f\left(x\right),\text{replace}x\text{by}\sqrt{x+1}\text{in}f\left(x\right)=x^2+4\\ ={\left(\sqrt{x+1}\right)}^2+4\\ =x+1+4\\ =x+5\end{array}$

Now, substitute 3 for x in $fog\left(x\right)=x+5$

  $\begin{array}{c}fog\left(3\right)=3+5\\ =8\end{array}$

Further, by the definition of $gof\left(x\right)$

  $\begin{array}{c}gof\left(x\right)=g\left(f\left(x\right)\right)\leftarrow \text{use}\text{definition}\text{of}f\left(x\right)\text{, rereplace}f\left(x\right)\text{by}f\left(x\right)=x^2+4\\ =g\left(x^2+4\right)\leftarrow \text{use}\text{definition}\text{of}g\left(x\right),\text{replace}x\text{by}\left(x^2+4\right)\text{in}g\left(x\right)=\sqrt{x+1}\\ =\sqrt{\left(x^2+4\right)+1}\\ =\sqrt{x^2+5}\end{array}$

Now, substitute -2 for x , $gof\left(x\right)=\sqrt{x^2+5}$ ,

  $\begin{array}{c}gof\left(-2\right)=\sqrt{{\left(-2\right)}^2+5}\\ =\sqrt{9}\\ =3\end{array}$

Therefore, $fog\left(3\right)=8$ and $gof\left(-2\right)=3$ .