4.7: Composite functions

Similar to the way in which we used transformations to analyze the equation of a function, it is sometimes helpful to consider a given function as being several functions of the variable combined together.

For example, instead of thinking of the function $f(x)=(2 x-7)^{3}$ as being a single function, we can think of it as being two functions:
$$\begin{array}{c} g(x)=2 x-7 \\ \text { and } \\ h(x)=x^{3} \end{array}$$
Then $f(x)$ is the combination or "composition" of these two functions together. The first function multiplies the variable by $2,$ and subtracts 7 from the result. The second function takes this answer and raises it to the third power. The notation for the composition of functions is an open circle: 0

In the example above we would say that the function $f(x)=(2 x-7)^{3}$ is equivalent to the composition $h \circ g(x)$ or $h(g(x))$. The order of function composition is important. The function $g \circ h(x)$ would be equivalent to $g(h(x)),$ which would be
$$\text { equal to } g\left(x^{3}\right)=2\left(x^{3}\right)-7=2 x^{3}-7$$

Exercises 4.7
Find $f \circ g(x)$ and $g \circ f(x)$ for each of the following problems.
1) $\quad f(x)=x^{2} \\ g(x)=x-1$
2) $\quad f(x)=|x-3| \\ g(x)=2 x+3$

3) $\quad f(x)=\frac{x}{x-2} \\ g(x)=\frac{x+3}{x}$
4) \(\quad f(x)=x^{3}-1 \\ g(x)=\frac{1}{x^{3}+1\)

5) $\quad f(x)=\sqrt{x+1} \\ g(x)=x^{4}-1$
6) \(\quad f(x)=2 x^{3}-1 \\ g(x)=\sqrt[3]{\frac{x+1}{2}\)

Find functions $f(x)$ and $g(x)$ so that the given function $h(x)=f \circ g(x)$
7) $\quad h(x)=(3 x+1)^{2}$
8) $\quad h(x)=\left(x^{2}-2 x\right)^{3}$
9) $\quad h(x)=\sqrt{1-4 x}$
10) $\quad h(x)=\sqrt[3]{x^{2}-1}$
11) $\quad h(x)=\left(\frac{x+1}{x-1}\right)^{2}$
12) $\quad h(x)=\left(\frac{1-2 x}{1+2 x}\right)^{3}$
13) $\quad h(x)=\left(3 x^{2}-1\right)^{-3}$
14) $\quad h(x)=\left(1+\frac{1}{x}\right)^{-2}$
15) $\quad h(x)=\sqrt{\frac{x}{x-1}}$
16) $\quad h(x)=\sqrt[3]{\frac{x-1}{x}}$
17) $\quad h(x)=\sqrt{\left(x^{2}-x-1\right)^{3}}$
18) $\quad h(x)=\sqrt[3]{\left(1-x^{4}\right)^{2}}$
19) $\quad h(x)=\frac{2}{\sqrt{4-x^{2}}}$
20) $\quad h(x)=-\left(\frac{3}{x-1}\right)^{5}$

21) A spherical weather balloon is inflated so that the radius at time $t$ is given by the equation:
$$r=f(t)=\frac{1}{2} t+2$$
Assume that $r$ is in meters and $t$ is in seconds, with $t=0$ corresponding to the time the balloon begins to be inflated. If the volume of a sphere is given by the formula:
$$v(r)=\frac{4}{3} \pi r^{3}$$
Find $V(f(t))$ and use this to compute the time at which the volume of the balloon is $36 \pi \mathrm{m}^{3}$