Chapter 1, Problem 16CT

a.

To determine

To find : the function and find domain.

Answer to Problem 16CT

The function $\left(f-g\right)\left(x\right)$ is $x^2-\sqrt{2-x}$ and the domain of the function is $x\le 2$.

Explanation of Solution

Given information :

The function are $f\left(x\right)=x^2\text{ and }g\left(x\right)=\sqrt{2-x}$

calculation : to find the function $\left(f-g\right)\left(x\right)$ subtract both the functions,

  $\left(f-g\right)\left(x\right)=x^2-\sqrt{2-x}$

The function has real values for these x values,

  $\begin{array}{l}2-x\ge 0\\ x\le 2\text{ [add both side }x]\\ \end{array}$

So the domain of the function is $x\le 2$.

b.

To determine

To find : the function and find domain.

Answer to Problem 16CT

The function $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{x^2}{\sqrt{2-x}}$ and the domain of the function is $x<2$.

Explanation of Solution

Given information :

The function are $f\left(x\right)=x^2\text{ and }g\left(x\right)=\sqrt{2-x}$

calculation : to find the function $\left(\frac{f}{g}\right)\left(x\right)$ divide both the functions,

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{x^2}{\sqrt{2-x}}$

The function has real values for these x values,

  $\begin{array}{l}2-x>0\\ x<2\text{ [add both side }x]\\ \end{array}$

So the domain of the function is $x<2$.

c.

To determine

To find : the function and find domain.

Answer to Problem 16CT

The function $\left(fog\right)\left(x\right)$ is $2-x$ and the domain of the function is $R$.

Explanation of Solution

Given information :

The function are $f\left(x\right)=x^2\text{ and }g\left(x\right)=\sqrt{2-x}$

calculation : to find the function $\left(fog\right)\left(x\right)$ substitute $x$ of function $f\left(x\right)$ by the function of $g\left(x\right)$

  $\begin{array}{l}\left(fog\right)\left(x\right)={\left(\sqrt{2-x}\right)}^2\\ \left(fog\right)\left(x\right)=2-x\text{ [square]}\end{array}$

The function is defined for all the real numbers.

So the domain of the function is $R$.

d.

To determine

To find : the function and find domain.

Answer to Problem 16CT

The function $\left(gof\right)\left(x\right)$ is $\sqrt{2-x^2}$ and the domain of the function is $x\le \sqrt{2}$.

Explanation of Solution

Given information :

The function are $f\left(x\right)=x^2\text{ and }g\left(x\right)=\sqrt{2-x}$

calculation : to find the function $\left(gof\right)\left(x\right)$ substitute $x$ of function $g\left(x\right)$ by the function of $f\left(x\right)$

  $\left(gof\right)\left(x\right)=\sqrt{2-x^2}$

The function is defined for the values of following as:

  $\begin{array}{l}2-x^2\ge 0\\ x\le \sqrt{2}\text{ [add and sqaure root]}\end{array}$

So the domain of the function is $x\le \sqrt{2}$.