Chapter 2.1, Problem 68AYU

a.

To determine

The value of $\left(f+g\right)\left(x\right)$ and the domain.

Answer to Problem 68AYU

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

Domain of $\left(f+g\right)\left(x\right)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Formula Used:

Let f and g be two functions.

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

For a composite function $f+g$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

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

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

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

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

As domain of $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ is set of real numbers.

So, domain of $f+g$ is set of all real numbers.

b.

To determine

The value of $\left(f-g\right)\left(x\right)$ and its domain.

Answer to Problem 68AYU

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

Domain of $\left(f-g\right)\left(x\right)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Formula Used:

Let f and g be two functions.

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

For a composite function $f-g$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

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

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

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

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

As domain of $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ is set of real numbers.

So, domain of $f-g$ is set of all real numbers.

c.

To determine

The value of $\left(fg\right)\left(x\right)$ and its domain.

Answer to Problem 68AYU

The value of $\left(fg\right)\left(x\right)$ is $\left|x\right|x$ .

Domain of $\left(fg\right)\left(x\right)$ is set of all real numbers.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Formula Used:

Let f and g be two functions.

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

For a composite function $fg$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

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

Substitute $\left|x\right|$ for $f\left(x\right)$ and $x$ for $g\left(x\right)$ i in the product formula.

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)\\ =\left|x\right|x\end{array}$

Hence, the value of $\left(fg\right)\left(x\right)$ is $\left|x\right|x$ .

As domain of $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ is set of real numbers.

So, domain of $\left(fg\right)\left(x\right)$ is set of all real numbers.

d.

To determine

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 68AYU

The value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left|x\right|}{x}$ and the domain is all the real numbers $x$ except $x=0$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Formula Used:

Let f and g be two functions.

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

For a composite function $f-g$ , the domain of the function is defined by

  $\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)$

If domain is not specified for a function, one should consider the set of largest real number where the function is defined all over.

Calculation:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

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

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

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

Find the domain of the function.

  $x=0$

The domain is all the real numbers $x$ except $x=0$ because the denominator becomes zero at $x=0$ .

Hence, the value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{\left|x\right|}{x}$ and the domain is all the real numbers $x$ except $x=0$ .

e)

To determine

The value of $\left(f+g\right)\left(3\right)$ .

Answer to Problem 68AYU

The value of $\left(f+g\right)\left(3\right)$ is $6$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Calculation:

Now, from part (a)

  $\left(f+g\right)\left(x\right)=\left|x\right|+x$

Substitute $3$ for $x$ in the composite function $\left(f+g\right)\left(x\right)=\left|x\right|+x$ .

  $\begin{array}{c}\left(f+g\right)\left(x\right)=\left|x\right|+x\\ \left(f+g\right)\left(3\right)=\left|3\right|+3\\ =3+3\\ =6\end{array}$

Hence, the value of $\left(f+g\right)\left(3\right)$ is $6$ .

f)

To determine

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

Answer to Problem 68AYU

The value of $\left(f-g\right)\left(4\right)$ is $0$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Calculation:

Now, from part (b)

  $\left(f-g\right)\left(x\right)=\left|x\right|-x$

Substitute $4$ for $x$ in the composite function $\left(f-g\right)\left(x\right)=\left|x\right|-x$ .

  $\begin{array}{c}\left(f-g\right)\left(x\right)=\left|x\right|-x\\ \left(f-g\right)\left(4\right)=\left|4\right|-4\\ =4-4\\ =0\end{array}$

Hence, the value of $\left(f-g\right)\left(4\right)$ is $0$ .

g)

To determine

The value of $\left(fg\right)\left(2\right)$ .

Answer to Problem 68AYU

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

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Calculation:

Now, from part (c)

  $\left(fg\right)\left(x\right)=\left|x\right|x$

Substitute $2$ for $x$ in the composite function $\left(fg\right)\left(x\right)=\left|x\right|x$ .

  $\begin{array}{c}\left(fg\right)\left(x\right)=\left|x\right|x\\ \left(fg\right)\left(2\right)=\left|2\right|2\\ =4\end{array}$

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

h)

To determine

The value of $\left(\frac{f}{g}\right)\left(1\right)$ .

Answer to Problem 68AYU

The value of $\left(\frac{f}{g}\right)\left(1\right)$ is $1$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\left|x\right|$ and $g\left(x\right)=x$ .

Calculation:

Now, from part (d)

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{\left|x\right|}{x}$

Substitute $1$ for $x$ in the composite function $\left(\frac{f}{g}\right)\left(x\right)=\frac{\left|x\right|}{x}$ .

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{\left|x\right|}{x}\\ \left(\frac{f}{g}\right)\left(1\right)=\frac{\left|1\right|}{1}\\ =1\end{array}$

Hence, the value of $\left(\frac{f}{g}\right)\left(1\right)$ is $1$ .