Chapter 1.8, Problem 7E

a.

To determine

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

Answer to Problem 7E

The value of $\left(f+g\right)\left(x\right)$ is $x^2+4x-5$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=x^2$ and $g\left(x\right)=4x-5$ .

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 $g\left(x\right)=4x-5$ .

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 $4x-5$ for $g\left(x\right)$ in the sum formula.

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =x^2+4x-5\end{array}$

Hence, the value of $\left(f+g\right)\left(x\right)$ is $x^2+4x-5$ .

b.

To determine

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

Answer to Problem 7E

The value of $\left(f-g\right)\left(x\right)$ is $x^2-4x+5$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=x^2$ and $g\left(x\right)=4x-5$ .

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 $g\left(x\right)=4x-5$ .

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 $4x-5$ for $g\left(x\right)$ in the difference formula.

  $\begin{array}{c}\left(f-g\right)\left(x\right)=f\left(x\right)-g\left(x\right)\\ =x^2-\left(4x-5\right)\\ =x^2-4x+5\end{array}$

Hence, the value of $\left(f-g\right)\left(x\right)$ is $x^2-4x+5$ .

c.

To determine

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

Answer to Problem 7E

The value of $\left(fg\right)\left(x\right)$ is $4x^3-5x^2$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=x^2$ and $g\left(x\right)=4x-5$ .

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 $g\left(x\right)=4x-5$ .

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 $4x-5$ for $g\left(x\right)$ in the product formula.

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)\\ =\left(x^2\right)\left(4x-5\right)\\ =4x^3-5x^2\end{array}$

Hence, the value of $\left(fg\right)\left(x\right)$ is $4x^3-5x^2$ .

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 7E

The value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{x^2}{\left(4x-5\right)}$ and the domain is all the real numbers $x$ except $x=\frac{5}{4}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=x^2$ and $g\left(x\right)=4x-5$ .

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 $g\left(x\right)=4x-5$ .

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 $4x-5$ for $g\left(x\right)$ in the division formula.

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{x^2}{\left(4x-5\right)}\end{array}$

Find the domain of the function.

  $\begin{array}{c}4x-5=0\\ x=\frac{5}{4}\end{array}$

The domain is all the real numbers $x$ except $x=\frac{5}{4}$ because the denominator becomes zero at $x=2$ .

Hence, the value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{x^2}{\left(4x-5\right)}$ and the domain is all the real numbers $x$ except $x=\frac{5}{4}$ .