# 4.7: Composite functions

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

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) $$\quad h(x)=\sqrt{x^{2}-1}$$
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) $$\quad h(x)=\sqrt{\frac{x-1}{x}}$$
17) $$\quad h(x)=\sqrt{\left(x^{2}-x-1\right)^{3}}$$
18) $$\quad h(x)=\sqrt{\left(1-x^{4}\right)^{2}}$$
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 $$36 \pi \mathrm{m}^{3}$$

4.7: Composite functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.