# 3.4 Composition of functions  (Page 4/9)

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The gravitational force on a planet a distance r from the sun is given by the function $G\left(r\right).$ The acceleration of a planet subjected to any force $F$ is given by the function $a\left(F\right).$ Form a meaningful composition of these two functions, and explain what it means.

A gravitational force is still a force, so $a\left(G\left(r\right)\right)$ makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but $G\left(a\left(F\right)\right)$ does not make sense.

## Evaluating composite functions

Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting input and then use the inner function’s output as the input for the outer function.

## Evaluating composite functions using tables

When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.

## Using a table to evaluate a composite function

Using [link] , evaluate $\text{\hspace{0.17em}}f\left(g\left(3\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(3\right)\right).$

$x$ $f\left(x\right)$ $g\left(x\right)$
1 6 3
2 8 5
3 3 2
4 1 7

To evaluate $\text{\hspace{0.17em}}f\left(g\left(3\right)\right),\text{\hspace{0.17em}}$ we start from the inside with the input value 3. We then evaluate the inside expression $\text{\hspace{0.17em}}g\left(3\right)\text{\hspace{0.17em}}$ using the table that defines the function $\text{\hspace{0.17em}}g:\text{\hspace{0.17em}}$ $g\left(3\right)=2.\text{\hspace{0.17em}}$ We can then use that result as the input to the function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}g\left(3\right)\text{\hspace{0.17em}}$ is replaced by 2 and we get $\text{\hspace{0.17em}}f\left(2\right).\text{\hspace{0.17em}}$ Then, using the table that defines the function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ we find that $\text{\hspace{0.17em}}f\left(2\right)=8.$

$\begin{array}{ccc}\hfill g\left(3\right)& =& 2\hfill \\ \hfill f\left(g\left(3\right)\right)& =& f\left(2\right)=8\hfill \end{array}$

To evaluate $\text{\hspace{0.17em}}g\left(f\left(3\right)\right),\text{\hspace{0.17em}}$ we first evaluate the inside expression $\text{\hspace{0.17em}}f\left(3\right)\text{\hspace{0.17em}}$ using the first table: $\text{\hspace{0.17em}}f\left(3\right)=3.\text{\hspace{0.17em}}$ Then, using the table for $\text{\hspace{0.17em}}g\text{,\hspace{0.17em}}$ we can evaluate

$g\left(f\left(3\right)\right)=g\left(3\right)=2$

[link] shows the composite functions $\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\circ f\text{\hspace{0.17em}}$ as tables.

 $x$ $g\left(x\right)$ $f\left(g\left(x\right)\right)$ $f\left(x\right)$ $g\left(f\left(x\right)\right)$ 3 2 8 3 2

Using [link] , evaluate $\text{\hspace{0.17em}}f\left(g\left(1\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(4\right)\right).$

$f\left(g\left(1\right)\right)=f\left(3\right)=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(4\right)\right)=g\left(1\right)=3$

## Evaluating composite functions using graphs

When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the $\text{\hspace{0.17em}}x\text{-}$ and $y\text{-}$ axes of the graphs.

Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.

1. Locate the given input to the inner function on the $\text{\hspace{0.17em}}x\text{-}$ axis of its graph.
2. Read off the output of the inner function from the $\text{\hspace{0.17em}}y\text{-}$ axis of its graph.
3. Locate the inner function output on the $\text{\hspace{0.17em}}x\text{-}$ axis of the graph of the outer function.
4. Read the output of the outer function from the $\text{\hspace{0.17em}}y\text{-}$ axis of its graph. This is the output of the composite function.

## Using a graph to evaluate a composite function

Using [link] , evaluate $\text{\hspace{0.17em}}f\left(g\left(1\right)\right).$

To evaluate $\text{\hspace{0.17em}}f\left(g\left(1\right)\right),\text{\hspace{0.17em}}$ we start with the inside evaluation. See [link] .

We evaluate $\text{\hspace{0.17em}}g\left(1\right)\text{\hspace{0.17em}}$ using the graph of $\text{\hspace{0.17em}}g\left(x\right),\text{\hspace{0.17em}}$ finding the input of 1 on the $\text{\hspace{0.17em}}x\text{-}$ axis and finding the output value of the graph at that input. Here, $\text{\hspace{0.17em}}g\left(1\right)=3.\text{\hspace{0.17em}}$ We use this value as the input to the function $\text{\hspace{0.17em}}f.$

$f\left(g\left(1\right)\right)=f\left(3\right)$

We can then evaluate the composite function by looking to the graph of $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ finding the input of 3 on the $x\text{-}$ axis and reading the output value of the graph at this input. Here, $\text{\hspace{0.17em}}f\left(3\right)=6,\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}f\left(g\left(1\right)\right)=6.$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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