3.4 Composition of functions  (Page 7/9)

What is the composition of two functions, $f\circ g?$

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.

How do you find the domain for the composition of two functions, $f\circ g?$

Given $f(x)=x^2+2x$ and $g(x)=6-x^2,$ find $f+g,f-g,fg,$ and $\frac{f}{g}.$

Given $f(x)=-3x^2+x$ and $g(x)=5,$ find $f+g,f-g,fg,$ and $\frac{f}{g}.$

Given $f(x)=2x^2+4x$ and $g(x)=\frac{1}{2x},$ find $f+g,f-g,fg,$ and $\frac{f}{g}.$

Given $f(x)=\frac{1}{x-4}$ and $g(x)=\frac{1}{6-x},$ find $f+g,f-g,fg,$ and $\frac{f}{g}.$

Given $f(x)=3x^2$ and $g(x)=\sqrt{x-5},$ find $f+g,f-g,fg,$ and $\frac{f}{g}.$

Given $f(x)=\sqrt{x}$ and $g(x)=|x-3|,$ find $\frac{g}{f}.$

For the following exercise, find the indicated function given $f(x)=2x^2+1$ and $g(x)=3x-5.$

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

$f(x)=x^2+1,g(x)=\sqrt{x+2}$

$f(x)=\sqrt{x}+2,g(x)=x^2+3$

$f(x)=\left|x\right|,g(x)=5x+1$

$f(x)=\sqrt[3]{x},g(x)=\frac{x+1}{x^3}$

$f(x)=\frac{1}{x-6},g(x)=\frac{7}{x}+6$

$f(x)=\frac{1}{x-4},g(x)=\frac{2}{x}+4$

$f(x)=x^4+6,$ $g(x)=x-6,$ and $h(x)=\sqrt{x}$

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

Given $f(x)=\frac{1}{x}$ and $g(x)=x-3,$ find the following:

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

Given $f(x)=\sqrt{2-4x}$ and $g(x)=-\frac{3}{x},$ find the following:

1. $(g\circ f)(x)$
2. the domain of $(g\circ f)(x)$ in interval notation

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

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

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

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

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

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

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

$h(x)={(x+2)}^2$

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

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

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

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

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

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

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