Chapter 1.8, Problem 5E

a.

To determine

Find the value of $\left(f+g\right)\left(x\right)$ .

Answer to Problem 5E

The value of $\left(f+g\right)\left(x\right)$ is $2x$ .

Explanation of Solution

Given information:

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

Formula Used:

Let f and g be two functions.

Use the sum formula, $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$ .

Calculation:

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

Use the sum formula, $\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)$ .

Substitute $x+2$ for $f\left(x\right)$ and $x-2$ for $g\left(x\right)$ in the sum formula.

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =x+2+x-2\\ =2x\end{array}$

Hence, the value of $\left(f+g\right)\left(x\right)$ is $2x$ .

b.

To determine

Find the value of $\left(f-g\right)\left(x\right)$ .

Answer to Problem 5E

The value of $\left(f-g\right)\left(x\right)$ is $4$ .

Explanation of Solution

Given information:

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

Formula Used:

Let f and g be two functions.

Use the difference formula, $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$ .

Calculation:

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

Use the difference formula, $\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)$ .

Substitute $x+2$ for $f\left(x\right)$ and $x-2$ for $g\left(x\right)$ in the difference formula.

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =x+2-x+2\\ =4\end{array}$

Hence, the value of $\left(f-g\right)\left(x\right)$ is $4$ .

c.

To determine

Find the value of $\left(fg\right)\left(x\right)$ .

Answer to Problem 5E

The value of $\left(fg\right)\left(x\right)$ is $x^2-4$ .

Explanation of Solution

Given information:

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

Formula Used:

Let f and g be two functions.

Use the product formula, $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$ .

Calculation:

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

Use the product formula, $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$ .

Substitute $x+2$ for $f\left(x\right)$ and $x-2$ for $g\left(x\right)$ in the product formula.

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)\\ =\left(x+2\right)\left(x-2\right)\\ =x^2-4\end{array}$

Hence, the value of $\left(fg\right)\left(x\right)$ is $x^2-4$ .

d.

To determine

Find the value of $\left(\frac{f}{g}\right)\left(x\right)$ . Also, find the domain of the function $\left(\frac{f}{g}\right)$ .

Answer to Problem 5E

The value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left(x+2\right)}{\left(x-2\right)}$ and the domain is all the real numbers $x$ except $x=2$ .

Explanation of Solution

Given information:

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

Formula Used:

Let f and g be two functions.

Use the division formula, $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ .

Calculation:

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

Use the division formula, $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ .

Substitute $x+2$ for $f\left(x\right)$ and $x-2$ for $g\left(x\right)$ in the division formula.

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{\left(x+2\right)}{\left(x-2\right)}\end{array}$

Find the domain of the function.

  $\begin{array}{c}x-2=0\\ x=2\end{array}$

The domain is all the real numbers $x$ except $x=2$ because the denominator becomes zero at $x=2$ .

Hence, the value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left(x+2\right)}{\left(x-2\right)}$ and the domain is all the real numbers $x$ except $x=2$ .