# 6.3: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Exercise $$\PageIndex{1}$$

Find $$f+g$$, $$f-g$$, $$f\cdot g$$ for the functions below. State their domain.

1. $$f(x)=x^2+6x$$, $$\quad \quad$$ and $$g(x)=3x-5$$
2. $$f(x)=x^3+5$$, $$\quad \quad$$ and $$g(x)=5x^2+7$$
3. $$f(x)=3x+7\sqrt{x}$$, $$\quad$$ and $$g(x)=2x^2+5\sqrt{x}$$
4. $$f(x)=\dfrac{1}{x+2}$$, $$\quad$$ and $$g(x)=\dfrac{5x}{x+2}$$
5. $$f(x)=\sqrt{x-3}$$, $$\quad \quad$$ and $$g(x)=2\sqrt{x-3}$$
6. $$f(x)=x^2+2x+5$$, $$\quad \quad$$ and $$g(x)=3x-6$$
7. $$f(x)=x^2+3x$$, $$\quad \quad$$ and $$g(x)=2x^2+3x+4$$
1. $$(f+g)(x)=x^{2}+9 x-5$$ with domain $$D_{f+g}=\mathbb{R},(f-g)(x)= x^2 + 3x + 5$$ with domain $$D_{f-g}=\mathbb{R},(f \cdot g)(x)=3 x^{3}+13 x^{2}-30 x$$ with domain $$D_{f \cdot g}=\mathbb{R}$$
2. $$(f+g)(x)=x^{3}+5 x^{2}+12, D_{f+g}=\mathbb{R},(f-g)(x)=x^{3}-5 x^{2}-2, D_{f-g}=\mathbb{R},(f \cdot g)(x)=5 x^{5}+7 x^{3}+25 x^{2}+35, D_{f \cdot g}=\mathbb{R}$$
3. $$(f+g)(x)=2 x^{2}+3 x+12 \sqrt{x}, D_{f+g}=[0, \infty),(f-g)(x)=-2 x^{2}+3 x+2 \sqrt{x}, D_{f-g}=[0, \infty),(f \cdot g)(x)=6 x^{3}+14 x^{2} \sqrt{x}+15 x \sqrt{x}+35 x, D_{f \cdot g}=[0, \infty)$$
4. $$(f+g)(x)=\dfrac{5 x+1}{x+2}, D_{f+g}=\mathbb{R}-\{-2\},(f-g)(x)=\dfrac{1-5 x}{x+2}, D_{f-g}=\mathbb{R}-\{-2\},(f \cdot g)(x)=\dfrac{5 x}{(x+2)^{2}}, D_{f \cdot g}=\mathbb{R}-\{-2\}$$
5. $$(f+g)(x)=3 \sqrt{x-3}, D_{f+g}=[3, \infty),(f-g)(x)=-\sqrt{x-3}, D_{f-g}=[3, \infty),(f \cdot g)(x)=2 \cdot(\sqrt{x-3})^{2}=2 \cdot(x-3), D_{f \cdot g}=[3, \infty)$$
6. $$(f+g)(x)=x^{2}+5 x-1, D_{f+g}=\mathbb{R},(f-g)(x)=x^{2}-x+11, D_{f-g}=\mathbb{R},(f \cdot g)(x)=3 x^{3}+3 x-30, D_{f \cdot g}=\mathbb{R}$$
7. $$(f+g)(x)=3 x^{2}+6 x+4, D_{f+g}=\mathbb{R},(f-g)(x)=-x^{2}-4, D_{f-g}=\mathbb{R},(f \cdot g)(x)=2 x^{4}+9 x^{3}+13 x^{2}+12 x, D_{f \cdot g}=\mathbb{R}$$

## Exercise $$\PageIndex{2}$$

Find $$\dfrac f g$$, and $$\dfrac g f$$ for the functions below. State their domain.

1. $$f(x)=3x+6$$, $$\quad \quad$$ and $$g(x)=2x-8$$
2. $$f(x)=x+2$$, $$\quad \quad$$ and $$g(x)=x^2-5x+4$$
3. $$f(x)=\dfrac{1}{x-5}$$, $$\quad \quad$$ and $$g(x)=\dfrac{x-2}{x+3}$$
4. $$f(x)=\sqrt{x+6}$$, $$\quad \quad$$ and $$g(x)=2x+5$$
5. $$f(x)=x^2+8x-33$$, $$\quad \quad$$ and $$g(x)=\sqrt{x}$$
1. $$\left(\dfrac{f}{g}\right)(x)=\dfrac{3 x+6}{2 x-8}$$ with domain $$D_{\frac{f}{g}}=\mathbb{R}-\{4\}$$, $$\left(\dfrac{g}{f}\right)(x)=\dfrac{2 x-8}{3 x+6}$$ with domain $$D_{\frac{g}{f}}=\mathbb{R}-\{-2\}$$
2. $$\left(\dfrac{f}{g}\right)(x)=\dfrac{x+2}{x^{2}-5 x+4}=\dfrac{x+2}{(x-4)(x-1)}$$, $$D_{\frac{f}{g}}=\mathbb{R}-\{1,4\}, \quad\left(\dfrac{g}{f}\right)(x)=\dfrac{x^{2}-5 x+4}{x+2}, D_{\frac{g}{f}}=\mathbb{R}-\{-2\}$$
3. $$\left(\dfrac{f}{g}\right)(x)=\dfrac{x+3}{(x-5)(x-2)}, D_{\frac{f}{g}}=\mathbb{R}-\{-3,2,5\},\left(\dfrac{g}{f}\right)(x)=\dfrac{(x-5)(x-2)}{x+3}, D_{\frac{g}{f}}=\mathbb{R}-\{-3,5\}$$
4. $$\left(\dfrac{f}{g}\right)(x)=\dfrac{\sqrt{x+6}}{2 x+5}, D_{\frac{f}{g}}=\left[-6,-\dfrac{5}{2}\right) \cup\left(-\dfrac{5}{2}, \infty\right), \left(\dfrac{g}{f}\right)(x)=\dfrac{2 x+5}{\sqrt{x+6}}, D_{\frac{g}{f}}=(-6, \infty)$$
5. $$\left(\dfrac{f}{g}\right)(x)=\dfrac{x^{2}+8 x-33}{\sqrt{x}}, D_{\frac{f}{g}}=(0, \infty), \left(\dfrac{g}{f}\right)(x)=\dfrac{\sqrt{x}}{x^{2}+8 x-33}, D_{\frac{g}{f}}=[0,3) \cup(3, \infty)$$

## Exercise $$\PageIndex{3}$$

Let $$f(x)=2x-3$$ and $$g(x)=3x^2+4x$$. Find the following compositions

1. $$f(g(2))$$
2. $$g(f(2))$$
3. $$f(f(5))$$
4. $$f(5 g(-3))$$
5. $$g(f(2)-2)$$
6. $$f(f(3)+g(3))$$
7. $$g(f(2+x))$$
8. $$f(f(-x))$$
9. $$f( f(-3)-3 g(2))$$
10. $$f(f(f(2)))$$
11. $$f(x+h)$$
12. $$g(x+h)$$
1. $$37$$
2. $$7$$
3. $$11$$
4. $$147$$
5. $$-1$$
6. $$81$$
7. $$12 x^{2}+20 x+7$$
8. $$-4 x-9$$
9. $$-141$$
10. $$-5$$
11. $$2 x+2 h-3$$
12. $$3 x^{2}+6 x h+3 h^{2}+4 x+4 h$$

## Exercise $$\PageIndex{4}$$

Find the composition $$(f\circ g)(x)$$ for the functions:

1. $$f(x)=3x-5$$, $$\quad \quad$$ and $$g(x)=2x+3$$
2. $$f(x)=x^2+2$$, $$\quad \quad$$ and $$g(x)=x+3$$
3. $$f(x)=x^2-3x+2$$, $$\quad \quad$$ and $$g(x)=2x+1$$
4. $$f(x)=x^2+\sqrt{x+3}$$, $$\quad \quad$$ and $$g(x)=x^2+2x$$
5. $$f(x)=\dfrac{2}{x+4}$$, $$\quad \quad$$ and $$g(x)=x+h$$
6. $$f(x)=x^2+4x+3$$, $$\quad \quad$$ and $$g(x)=x+h$$
1. $$(f \circ g)(x)=6 x+4$$
2. $$(f \circ g)(x)=x^{2}+6 x+11$$
3. $$(f \circ g)(x)=4 x^{2}-2x$$
4. $$(f \circ g)(x)=x^{4}+4 x^{3}+4 x^{2}+\sqrt{x^{2}+2 x+3}$$
5. $$(f \circ g)(x)=\dfrac{2}{x+h+4}$$
6. $$(f \circ g)(x)=x^{2}+2 x h+h^{2}+4 x+4 h+3$$

## Exercise $$\PageIndex{5}$$

Find the compositions

$(f\circ g)(x),\quad (g\circ f)(x),\quad(f\circ f)(x),\quad(g\circ g)(x) \nonumber$

for the following functions:

1. $$f(x)=2x+4$$, $$\quad \quad$$ and $$g(x)=x-5$$
2. $$f(x)=x+3$$, $$\quad \quad$$ and $$g(x)=x^2-2x$$
3. $$f(x)=2x^2-x-6$$, $$\quad \quad$$ and $$g(x)=\sqrt{3x+2}$$
4. $$f(x)=\dfrac{1}{x+3}$$, $$\quad \quad$$ and $$g(x)=\dfrac{1}{x}-3$$
5. $$f(x)=(2x-7)^2$$, $$\quad \quad$$ and $$g(x)=\dfrac{\sqrt{x}+7}{2}$$
1. $$(f \circ g)(x)=2 x-6, (g \circ f)(x)=2 x-1, (f \circ f)(x)=4 x+12, (g \circ g)(x)=x-10$$
2. $$(f \circ g)(x)=x^{2}-2 x+3, (g \circ f)(x)=x^{2}+4 x+3, (f \circ f)(x)=x+6, (g \circ g)(x)=x^{4}-4 x^{3}+2 x^{2}+4 x$$
3. $$(f \circ g)(x)=6 x-2-\sqrt{3 x+2}, (g \circ f)(x)=\sqrt{6 x^{2}-3 x-16}, (f \circ f)(x)=8 x^{4}-8 x^{3}-48 x^{2}+25 x+72, (g \circ g)(x)=\sqrt{3 \sqrt{3 x+2}+2}$$
4. $$(f \circ g)(x)=x, (g \circ f)(x)=x, (f \circ f)(x)=\dfrac{x+3}{3 x+10}, (g \circ g)(x)=\dfrac{10 x-3}{1-3 x}$$
5. $$(f \circ g)(x)=x,(g \circ f)(x)=x,(f \circ f)(x)=\left(2(2 x-7)^{2}-7\right)^{2}$$ or expanded in descending degrees: $$(f \circ f)(x)=64 x^{4}-896 x^{3}+4592 x^{2}-10192 x+8281,(g \circ g)(x)=\dfrac{\sqrt{\frac{\sqrt{x}+7}{2}}+7}{2}=\dfrac{14+\sqrt{14+2 \sqrt{x}}}{4}$$

## Exercise $$\PageIndex{6}$$

Let $$f$$ and $$g$$ be the functions defined by the following table. Complete the table given below.

$\begin{array}{|c||c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \hline f(x) & 4 & 5 & 7 & 0 & -2 & 6 & 4 \\ \hline g(x) & 6 & -8 & 5 & 2 & 9 & 11 & 2 \\ \hline f(x)+3 & & & & & & & \\ \hline 4 g(x)+5 & & & & & & & \\ \hline g(x)-2 f(x) & & & & & & & \\ \hline f(x+3) & & & & & & & \\ \hline \end{array} \nonumber$

$$\begin{array}{|c||c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \hline f(x) & 4 & 5 & 7 & 0 & -2 & 6 & 4 \\ \hline g(x) & 6 & -8 & 5 & 2 & 9 & 11 & 2 \\ \hline f(x)+3 & 7 & 8 & 10 & 3 & 1 & 9 & 7 \\ \hline 4 g(x)+5 & 29 & -28 & 25 & 13 & 41 & 49 & 13 \\ \hline g(x)-2 f(x) & -2 & -18 & -9 & 2 & 13 & -1 & -6 \\ \hline f(x+3) & 0 & -2 & 6 & 4 & \text { undef. } & \text { undef. } & \text { undef. } \\ \hline \end{array} \nonumber$$

Note, however, that the complete table for $$y = f(x + 3)$$ is given by:

$$\begin{array}{|c||c|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline \hline f(x+3) & 4 & 5 & 7 & 0 & -2 & 6 & 4 \\ \hline \end{array} \nonumber$$

## Exercise $$\PageIndex{7}$$

Let $$f$$ and $$g$$ be the functions defined by the following table. Complete the table by composing the given functions.

$\begin{array}{|c||c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \hline f(x) & 3 & 1 & 2 & 5 & 6 & 3 \\ \hline g(x) & 5 & 2 & 6 & 1 & 2 & 4 \\ \hline(g \circ f)(x) & & & & & & \\ \hline(f \circ g)(x) & & & & & & \\ \hline(f \circ f)(x) & & & & & & \\ \hline(g \circ g)(x) & & & & & & \\ \hline \end{array} \nonumber$

$$\begin{array}{|c||c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \hline f(x) & 3 & 1 & 2 & 5 & 6 & 3 \\ \hline g(x) & 5 & 2 & 6 & 1 & 2 & 4 \\ \hline(g \circ f)(x) & 6 & 5 & 2 & 2 & 4 & 6 \\ \hline(f \circ g)(x) & 6 & 1 & 3 & 3 & 1 & 5 \\ \hline(f \circ f)(x) & 2 & 3 & 1 & 6 & 3 & 2 \\ \hline(g \circ g)(x) & 2 & 2 & 4 & 5 & 2 & 1 \\ \hline \end{array} \nonumber$$

## Exercise $$\PageIndex{8}$$

Let $$f$$ and $$g$$ be the functions defined by the following table. Complete the table by composing the given functions.

$\begin{array}{|c||c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \hline f(x) & 4 & 8 & 5 & 6 & 12 & -1 & 10 \\ \hline g(x) & 10 & 2 & 0 & -6 & 7 & 2 & 8 \\ \hline(g \circ f)(x) & & & & & & & \\ \hline(f \circ g)(x) & & & & & & & \\ \hline(f \circ f)(x) & & & & & & & \\ \hline(g \circ g)(x) & & & & & & & \\ \hline \end{array} \nonumber$

$$\begin{array}{|c||c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \hline f(x) & 4 & 8 & 5 & 6 & 12 & -1 & 10 \\ \hline g(x) & 10 & 2 & 0 & -6 & 7 & 2 & 8 \\ \hline(g \circ f)(x) & 0 & 7 & \text { undef. } & -6 & 8 & \text { undef. } & 2 \\ \hline(f \circ g)(x) & -1 & 8 & 4 & \text { undef. } & \text { undef. } & 8 & 12 \\ \hline(f \circ f)(x) & 5 & 12 & \text { undef. } & 6 & 10 & \text { undef. } & -1 \\ \hline(g \circ g)(x) & 2 & 2 & 10 & \text { undef. } & \text { undef. } & 2 & 7 \\ \hline \end{array} \nonumber$$