# 3.4 Composition of functions  (Page 7/9)

 Page 7 / 9

What is the composition of two functions, $\text{\hspace{0.17em}}f\circ g?$

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

Yes. Sample answer: Let Then $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=f\left(x-1\right)=\left(x-1\right)+1=x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=g\left(x+1\right)=\left(x+1\right)-1=x.\text{\hspace{0.17em}}$ So $\text{\hspace{0.17em}}f\circ g=g\circ f.$

How do you find the domain for the composition of two functions, $\text{\hspace{0.17em}}f\circ g?$

## Algebraic

For the following exercises, determine the domain for each function in interval notation.

Given and find and

$\left(f+g\right)\left(x\right)=2x+6,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(f-g\right)\left(x\right)=2{x}^{2}+2x-6,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(fg\right)\left(x\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{{x}^{2}+2x}{6-{x}^{2}},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,-\sqrt{6}\right)\cup \left(-\sqrt{6},\sqrt{6}\right)\cup \left(\sqrt{6},\infty \right)$

Given and find $\text{\hspace{0.17em}}f+g,\text{\hspace{0.17em}}f-g,\text{\hspace{0.17em}}fg,\text{\hspace{0.17em}}$ and

Given and find and

$\left(f+g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}+1}{2x},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(f-g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}-1}{2x},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(fg\right)\left(x\right)=x+2,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=4{x}^{3}+8{x}^{2},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Given $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x-4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{6-x},\text{\hspace{0.17em}}$ find and

Given $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-5},\text{\hspace{0.17em}}$ find and

$\left(f+g\right)\left(x\right)=3{x}^{2}+\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(f-g\right)\left(x\right)=3{x}^{2}-\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(fg\right)\left(x\right)=3{x}^{2}\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{3{x}^{2}}{\sqrt{x-5}},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(5,\infty \right)$

Given $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=|x-3|,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\frac{g}{f}.\text{\hspace{0.17em}}$

For the following exercise, find the indicated function given $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=3x-5.\text{\hspace{0.17em}}$

1. $f\left(g\left(2\right)\right)$
2. $f\left(g\left(x\right)\right)$
3. $g\left(f\left(x\right)\right)$
4. $\left(g\circ g\right)\left(x\right)$
5. $\left(f\circ f\right)\left(-2\right)$

a. 3; b. $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=2{\left(3x-5\right)}^{2}+1;\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=6{x}^{2}-2;\text{\hspace{0.17em}}$ d. $\text{\hspace{0.17em}}\left(g\circ g\right)\left(x\right)=3\left(3x-5\right)-5=9x-20;\text{\hspace{0.17em}}$ e. $\text{\hspace{0.17em}}\left(f\circ f\right)\left(-2\right)=163$

For the following exercises, use each pair of functions to find $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(x\right)\right).\text{\hspace{0.17em}}$ Simplify your answers.

$f\left(x\right)={x}^{2}+1,\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x+2}$

$f\left(x\right)=\sqrt{x}+2,\text{\hspace{0.17em}}g\left(x\right)={x}^{2}+3$

$f\left(g\left(x\right)\right)=\sqrt{{x}^{2}+3}+2,\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x+4\sqrt{x}+7$

$f\left(x\right)=|x|,\text{\hspace{0.17em}}g\left(x\right)=5x+1$

$f\left(x\right)=\sqrt[3]{x},\text{\hspace{0.17em}}g\left(x\right)=\frac{x+1}{{x}^{3}}$

$f\left(g\left(x\right)\right)=\sqrt[3]{\frac{x+1}{{x}^{3}}}=\frac{\sqrt[3]{x+1}}{x},\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=\frac{\sqrt[3]{x}+1}{x}$

$f\left(x\right)=\frac{1}{x-6},\text{\hspace{0.17em}}g\left(x\right)=\frac{7}{x}+6$

$f\left(x\right)=\frac{1}{x-4},\text{\hspace{0.17em}}g\left(x\right)=\frac{2}{x}+4$

For the following exercises, use each set of functions to find $\text{\hspace{0.17em}}f\left(g\left(h\left(x\right)\right)\right).\text{\hspace{0.17em}}$ Simplify your answers.

$f\left(x\right)={x}^{4}+6,\text{\hspace{0.17em}}$ $g\left(x\right)=x-6,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=\sqrt{x}$

$f\left(x\right)={x}^{2}+1,\text{\hspace{0.17em}}$ $g\left(x\right)=\frac{1}{x},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=x+3$

$f\left(g\left(h\left(x\right)\right)\right)={\left(\frac{1}{x+3}\right)}^{2}+1$

Given $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=x-3,\text{\hspace{0.17em}}$ find the following:

1. $\left(f\circ g\right)\left(x\right)$
2. the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation
3. $\left(g\circ f\right)\left(x\right)$
4. the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)\text{\hspace{0.17em}}$
5. $\left(\frac{f}{g}\right)x$

Given $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{2-4x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=-\frac{3}{x},\text{\hspace{0.17em}}$ find the following:

1. $\left(g\circ f\right)\left(x\right)$
2. the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation

a. $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=-\frac{3}{\sqrt{2-4x}};\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(-\infty ,\frac{1}{2}\right)$

Given the functions $\text{\hspace{0.17em}}f\left(x\right)=\frac{1-x}{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{1+{x}^{2}},$ find the following:

1. $\left(g\circ f\right)\left(x\right)$
2. $\left(g\circ f\right)\left(\text{2}\right)$

Given functions $\text{\hspace{0.17em}}p\left(x\right)=\frac{1}{\sqrt{x}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}m\left(x\right)={x}^{2}-4,\text{\hspace{0.17em}}$ state the domain of each of the following functions using interval notation:

1. $\frac{p\left(x\right)}{m\left(x\right)}$
2. $p\left(m\left(x\right)\right)$
3. $m\left(p\left(x\right)\right)$

a. $\text{\hspace{0.17em}}\left(0,2\right)\cup \left(2,\infty \right);\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(-\infty ,-2\right)\cup \left(2,\infty \right);\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}\left(0,\infty \right)$

Given functions $\text{\hspace{0.17em}}q\left(x\right)=\frac{1}{\sqrt{x}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)={x}^{2}-9,\text{\hspace{0.17em}}$ state the domain of each of the following functions using interval notation.

1. $\frac{q\left(x\right)}{h\left(x\right)}$
2. $q\left(h\left(x\right)\right)$
3. $h\left(q\left(x\right)\right)$

For $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-1},\text{\hspace{0.17em}}$ write the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation.

$\left(1,\infty \right)$

For the following exercises, find functions $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ so the given function can be expressed as $\text{\hspace{0.17em}}h\left(x\right)=f\left(g\left(x\right)\right).$

$h\left(x\right)={\left(x+2\right)}^{2}$

$h\left(x\right)={\left(x-5\right)}^{3}$

sample: $\begin{array}{l}f\left(x\right)={x}^{3}\\ g\left(x\right)=x-5\end{array}$

$h\left(x\right)=\frac{3}{x-5}$

$h\left(x\right)=\frac{4}{{\left(x+2\right)}^{2}}$

sample: $\begin{array}{l}f\left(x\right)=\frac{4}{x}\hfill \\ g\left(x\right)={\left(x+2\right)}^{2}\hfill \end{array}$

$h\left(x\right)=4+\sqrt[3]{x}$

$h\left(x\right)=\sqrt[3]{\frac{1}{2x-3}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[3]{x}\\ g\left(x\right)=\frac{1}{2x-3}\end{array}$

$h\left(x\right)=\frac{1}{{\left(3{x}^{2}-4\right)}^{-3}}$

$h\left(x\right)=\sqrt[4]{\frac{3x-2}{x+5}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[4]{x}\\ g\left(x\right)=\frac{3x-2}{x+5}\end{array}$

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