4.7: Composite functions
- Page ID
- 40920
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(36 \pi \mathrm{m}^{3}\)