# 1.4 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 $\text{\hspace{0.17em}}f\left(2\right)=8.$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\left(3\right)=2\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\left(g\left(3\right)\right)=f\left(2\right)=8\hfill \end{array}$

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 $\text{\hspace{0.17em}}f\left(g\left(1\right)\right)=6.$

give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena