1.4 Composition of functions (Page 4/9)

Precalculus

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

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

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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 $f (g (3))$ and $g (f (3)) .$

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

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

\begin{array}{l} g (3) = 2 \\ f (g (3)) = f (2) = 8 \end{array}

To evaluate $g (f (3)),$ we first evaluate the inside expression $f (3)$ using the first table: $f (3) = 3.$ Then, using the table for $g,$ we can evaluate

g (f (3)) = g (3) = 2

[link] shows the composite functions $f \circ g$ and $g \circ f$ as tables.

$x$	$g (x)$	$f (g (x))$	$f (x)$	$g (f (x))$
3	2	8	3	2

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Try It

Using [link] , evaluate $f (g (1))$ and $g (f (4)) .$

$f (g (1)) = f (3) = 3$ and $g (f (4)) = g (1) = 3$

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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 $x -$ and $y -$ axes of the graphs.

How To

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

Locate the given input to the inner function on the $x -$ axis of its graph.
Read off the output of the inner function from the $y -$ axis of its graph.
Locate the inner function output on the $x -$ axis of the graph of the outer function.
Read the output of the outer function from the $y -$ axis of its graph. This is the output of the composite function.

Using a graph to evaluate a composite function

Using [link] , evaluate $f (g (1)) .$

To evaluate $f (g (1)),$ we start with the inside evaluation. See [link] .

Two graphs of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.

We evaluate $g (1)$ using the graph of $g (x),$ finding the input of 1 on the $x -$ axis and finding the output value of the graph at that input. Here, $g (1) = 3.$ We use this value as the input to the function $f .$

f (g (1)) = f (3)

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

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?

ADNAN Reply

linear speed of an object

Melissa Reply

an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object

Melissa

test

Matrix

how to find domain

Mohamed Reply

like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.

Dan

define the term of domain

Moha

if a>0 then the graph is concave

Angel Reply

if a<0 then the graph is concave blank

Angel

what's a domain

Kamogelo Reply

The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.

Spiro

how fast can i understand functions without much difficulty

Joe Reply

what is inequalities

Nathaniel

functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2

Dan

what is set?

Kelvin Reply

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size

Divya Reply

I got 300 minutes. is it right?

Patience

no. should be about 150 minutes.

Jason

It should be 158.5 minutes.

ok, thanks

Patience

100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes

Thomas

158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49

Dan

what is the importance knowing the graph of circular functions?

Arabella Reply

can get some help basic precalculus

ismail Reply

What do you need help with?

Andrew

how to convert general to standard form with not perfect trinomial

Camalia Reply

can get some help inverse function

ismail

Rectangle coordinate

Asma Reply

how to find for x

Jhon Reply

it depends on the equation

Robert

yeah, it does. why do we attempt to gain all of them one side or the other?

Melissa

how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2

Dan

whats a domain

mike Reply

The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.

Spiro

Spiro; thanks for putting it out there like that, 😁

Melissa

foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.

Churlene Reply

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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