Chapter 2.1, Problem 71AYU

a.

To determine

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

Answer to Problem 71AYU

The value of $\left(f+g\right)\left(x\right)$ is $\frac{6x+3}{3x-2}$ .

Domain of $\left(f+g\right)\left(x\right)$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Formula Used:

Let f and g be two functions.

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

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 $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

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

Substitute $\frac{2x+3}{3x-2}$ for $f\left(x\right)$ and $\frac{4x}{3x-2}$ for $g\left(x\right)$ in the sum formula.

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =\frac{2x+3}{3x-2}+\frac{4x}{3x-2}\\ =\frac{2x+3+4x}{3x-2}\\ =\frac{6x+3}{3x-2}\end{array}$

Hence, the value of $\left(f+g\right)\left(x\right)$ is $\frac{6x+3}{3x-2}$ .

Now, check the domain of $f\left(x\right)=\frac{2x+3}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $f\left(x\right)=\frac{2x+3}{3x-2}$ is set of all real numbersexcept $x=\frac{2}{3}$ .

Now, check the domain of $g\left(x\right)=\frac{4x}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $g\left(x\right)=\frac{4x}{3x-2}$ is set of all real numbers except $x=\frac{2}{3}$ .

So, domain of $f+g$ can be found by

. $\begin{array}{c}\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)=\left\{x|x\ne \frac{2}{3}\right\}\cap \left\{x|x\ne \frac{2}{3}\right\}\\ =\left\{x|x\ne \frac{2}{3}\right\}\end{array}$

So, domain of $f+g$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

b.

To determine

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

Answer to Problem 71AYU

The value of $\left(f-g\right)\left(x\right)$ is $\frac{3-2x}{3x-2}$ .

Domain of $\left(f-g\right)\left(x\right)$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Formula Used:

Let f and g be two functions.

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

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 $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

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

Substitute $\frac{2x+3}{3x-2}$ for $f\left(x\right)$ and $\frac{4x}{3x-2}$ for $g\left(x\right)$ in the difference formula.

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

Hence, the value of $\left(f-g\right)\left(x\right)$ is $\frac{3-2x}{3x-2}$ .

Now, check the domain of $f\left(x\right)=\frac{2x+3}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $f\left(x\right)=\frac{2x+3}{3x-2}$ is set of all real numbers except $x=\frac{2}{3}$ .

Now, check the domain of $g\left(x\right)=\frac{4x}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $g\left(x\right)=\frac{4x}{3x-2}$ is set of all real numbers except $x=\frac{2}{3}$ .

So, domain of $f-g$ can be found by

. $\begin{array}{c}\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)=\left\{x|x\ne \frac{2}{3}\right\}\cap \left\{x|x\ne \frac{2}{3}\right\}\\ =\left\{x|x\ne \frac{2}{3}\right\}\end{array}$

So, domain of $\left(f-g\right)\left(x\right)$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

c.

To determine

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

Answer to Problem 71AYU

The value of $\left(fg\right)\left(x\right)$ is $\frac{4x\left(2x+3\right)}{{\left(3x-2\right)}^2}$ .

Domain of $\left(fg\right)\left(x\right)$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Formula Used:

Let f and g be two functions.

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

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 $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

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

Substitute $\frac{2x+3}{3x-2}$ for $f\left(x\right)$ and $\frac{4x}{3x-2}$ for $g\left(x\right)$ in the product formula.

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

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

Now, check the domain of $f\left(x\right)=\frac{2x+3}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $f\left(x\right)=\frac{2x+3}{3x-2}$ is set of all real numbers except $x=\frac{2}{3}$ .

Now, check the domain of $g\left(x\right)=\frac{4x}{3x-2}$

  $\begin{array}{c}3x-2=0\\ x=\frac{2}{3}\end{array}$

Domain of $g\left(x\right)=\frac{4x}{3x-2}$ is set of all real numbers except $x=\frac{2}{3}$ .

So, domain of $\left(fg\right)\left(x\right)$ can be found by

. $\begin{array}{c}\left(\text{Domain}\text{of}f\right)\cap \left(\text{Domain}\text{of}g\right)=\left\{x|x\ne \frac{2}{3}\right\}\cap \left\{x|x\ne \frac{2}{3}\right\}\\ =\left\{x|x\ne \frac{2}{3}\right\}\end{array}$

So, domain of $\left(fg\right)\left(x\right)$ is $\left\{x|x\ne \frac{2}{3}\right\}$ .

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 71AYU

The value of $\left(\frac{f}{g}\right)\left(x\right)$ is $\frac{2x+3}{4x}$ and the domain is $\left\{x|x\ne 0\right\}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Formula Used:

Let f and g be two functions.

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

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 $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

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

Substitute $\frac{2x+3}{3x-2}$ for $f\left(x\right)$ and $\frac{4x}{3x-2}$ for $g\left(x\right)$ in the division formula.

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

Now, check the domain of $\left(\frac{f}{g}\right)\left(x\right)=\frac{2x+3}{4x}$

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

So, domain of $\left(\frac{f}{g}\right)$ is $\left\{x|x\ne 0\right\}$ .

e)

To determine

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

Answer to Problem 71AYU

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

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Calculation:

Now, from part (a)

  $\left(f+g\right)\left(x\right)=\frac{6x+3}{3x-2}$

Substitute $3$ for $x$ in the composite function $\left(f+g\right)\left(x\right)=\frac{6x+3}{3x-2}$ .

  $\begin{array}{c}\left(f+g\right)\left(x\right)=\frac{6x+3}{3x-2}\\ \left(f+g\right)\left(3\right)=\frac{6\cdot 3+3}{3\cdot 3-2}\\ =\frac{21}{7}\\ =3\end{array}$

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

f)

To determine

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

Answer to Problem 71AYU

The value of $\left(f-g\right)\left(4\right)$ is $-\frac{1}{2}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Calculation:

Now, from part (b)

  $\left(f-g\right)\left(x\right)=\frac{3-2x}{3x-2}$

Substitute $4$ for $x$ in the composite function $\left(f-g\right)\left(x\right)=\frac{3-2x}{3x-2}$ .

  $\begin{array}{c}\left(f-g\right)\left(x\right)=\frac{3-2x}{3x-2}\\ \left(f-g\right)\left(4\right)=\frac{3-2\cdot 4}{3\cdot 4-2}\\ =\frac{-5}{10}\\ =-\frac{1}{2}\end{array}$

Hence, the value of $\left(f-g\right)\left(4\right)$ is $-\frac{1}{2}$ .

g)

To determine

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

Answer to Problem 71AYU

The value of $\left(fg\right)\left(2\right)$ is $\frac{7}{2}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Calculation:

Now, from part (c)

  $\left(fg\right)\left(x\right)=\frac{4x\left(2x+3\right)}{{\left(3x-2\right)}^2}$

Substitute $2$ for $x$ in the composite function $\left(fg\right)\left(x\right)=\frac{4x\left(2x+3\right)}{{\left(3x-2\right)}^2}$ .

  $\begin{array}{c}\left(fg\right)\left(x\right)=\frac{4x\left(2x+3\right)}{{\left(3x-2\right)}^2}\\ \left(fg\right)\left(2\right)=\frac{4\cdot 2\left(2\cdot 2+3\right)}{{\left(3\cdot 2-2\right)}^2}\\ =\frac{8\cdot 7}{16}\\ =\frac{7}{2}\end{array}$

Hence, the value of $\left(fg\right)\left(2\right)$ is $\frac{7}{2}$ .

h)

To determine

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

Answer to Problem 71AYU

The value of $\left(\frac{f}{g}\right)\left(1\right)$ is $\frac{5}{4}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\frac{2x+3}{3x-2}$ and $g\left(x\right)=\frac{4x}{3x-2}$ .

Calculation:

Now, from part (d)

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{2x+3}{4x}$

Substitute $1$ for $x$ in the composite function $\left(\frac{f}{g}\right)\left(x\right)=\frac{2x+3}{4x}$ .

  $\begin{array}{c}\left(\frac{f}{g}\right)\left(x\right)=\frac{2x+3}{4x}\\ \left(\frac{f}{g}\right)\left(1\right)=\frac{2\cdot 1+3}{4\cdot 1}\\ =\frac{5}{4}\end{array}$

Hence, the value of $\left(\frac{f}{g}\right)\left(1\right)$ is $\frac{5}{4}$ .