6.3: Exercises

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 $g(x)=2x^2+3x+4$

Answer

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 $g(x)=\sqrt{x}$

Answer

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. $g(x+h)$

Answer

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 $g(x)=x+h$

Answer

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 $g(x)=\dfrac{\sqrt{x}+7}{2}$

Answer

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.

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

Answer

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

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.

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

Answer

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

Exercise $\PageIndex{8}$

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

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

Answer

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