Chapter 12, Problem 6CR

(a)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The value of $\left(f\circ g\right)\left(2\right)$ is $5$ .

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=f\left[2x+1\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{3\left(2x+1\right)}{2x+1-2}\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{6x+3}{2x-1}\\ \Rightarrow \left(f\circ g\right)\left(2\right)=\frac{6\times 2+3}{2\times 2-1}\left[\text{put }x=2\right]\\ \Rightarrow \left(f\circ g\right)\left(2\right)=\frac{12+3}{4-1}\\ \Rightarrow \left(f\circ g\right)\left(2\right)=\frac{15}{3}\\ \Rightarrow \left(f\circ g\right)\left(2\right)=5\end{array}$

Hence, the value of $\left(f\circ g\right)\left(2\right)$ is $5$ .

(b)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The value of $\left(g\circ f\right)\left(4\right)$ is $13$ .

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=g\left[\frac{3x}{x-2}\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=2\left(\frac{3x}{x-2}\right)+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x}{x-2}+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x+x-2}{x-2}\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{7x-2}{x-2}\\ \Rightarrow \left(g\circ f\right)\left(4\right)=\frac{7\times 4-2}{4-2}\left[\text{put }x=4\right]\\ \Rightarrow \left(g\circ f\right)\left(4\right)=\frac{28-2}{4-2}\\ \Rightarrow \left(g\circ f\right)\left(4\right)=\frac{26}{2}\\ \Rightarrow \left(g\circ f\right)\left(4\right)=13\end{array}$

Hence, the value of $\left(g\circ f\right)\left(4\right)$ is $13$ .

(c)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The value of $\left(f\circ g\right)\left(x\right)$ is $\frac{6x+3}{2x-1}$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=f\left[2x+1\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{3\left(2x+1\right)}{2x+1-2}\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{6x+3}{2x-1}\end{array}$

Hence, the value of $\left(f\circ g\right)\left(x\right)$ is $\frac{6x+3}{2x-1}$

(d)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The domain of $\left(f\circ g\right)\left(x\right)=\frac{6x+3}{2x-1}$ is $\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left[g\left(x\right)\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=f\left[2x+1\right]\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{3\left(2x+1\right)}{2x+1-2}\\ \Rightarrow \left(f\circ g\right)\left(x\right)=\frac{6x+3}{2x-1}\end{array}$

Hence, the domain of $\left(f\circ g\right)\left(x\right)=\frac{6x+3}{2x-1}$ is $\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$

(e)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The value of $\left(g\circ f\right)\left(x\right)$ is $\frac{7x-2}{x-2}$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=g\left[\frac{3x}{x-2}\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=2\left(\frac{3x}{x-2}\right)+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x}{x-2}+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x+x-2}{x-2}\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{7x-2}{x-2}\end{array}$

Hence, the value of $\left(g\circ f\right)\left(x\right)$ is $\frac{7x-2}{x-2}$

(f)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The domain is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left[f\left(x\right)\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=g\left[\frac{3x}{x-2}\right]\\ \Rightarrow \left(g\circ f\right)\left(x\right)=2\left(\frac{3x}{x-2}\right)+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x}{x-2}+1\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{6x+x-2}{x-2}\\ \Rightarrow \left(g\circ f\right)\left(x\right)=\frac{7x-2}{x-2}\end{array}$

Hence, the domain is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$

(g)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The domain of $g^{-1}\left(x\right)$ is $\left(-\infty ,\infty \right)$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

The function $g^{-1}$ and its domain

Here, $g\left(x\right)=2x+1$

To find the inverse, interchange the variable and solve for $y$ .

  $\begin{array}{l}\therefore g^{-1}\left(x\right)=\frac{x}{2}-\frac{1}{2}\\ \Rightarrow g^{-1}\left(x\right)=\frac{x-1}{2}\end{array}$

Hence, the domain of $g^{-1}\left(x\right)$ is $\left(-\infty ,\infty \right)$

(h)

To determine

To find:

1. $\left(f\circ g\right)\left(2\right)$
2. $\left(g\circ f\right)\left(4\right)$
3. $\left(f\circ g\right)\left(x\right)$
4. The domain of $\left(f\circ g\right)\left(x\right)$
5. $\left(g\circ f\right)\left(x\right)$
6. The domain of $\left(g\circ f\right)\left(x\right)$
7. The function $g^{-1}$ and its domain
8. The function $f^{-1}$ and its domain

Answer to Problem 6CR

The domain of $f^{-1}\left(x\right)$ is $\left(-\infty ,\infty \right)$

Explanation of Solution

Given information:

  $f\left(x\right)=\frac{3x}{x-2},$

  $g\left(x\right)=2x+1;$

Calculation:

The function $f^{-1}$ and its domain

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

To find the inverse, interchange the variable and solve for $y$ .

  $\therefore f^{-1}\left(x\right)=\frac{2x}{x-3}$

Hence, the domain of $f^{-1}\left(x\right)$ is $\left(-\infty ,\infty \right)$