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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.

g ( 3 ) = 2 f ( g ( 3 ) ) = f ( 2 ) = 8

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 g and g f as tables.

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

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 x - axis of its graph.
  2. Read off the output of the inner function from the y - axis of its graph.
  3. Locate the inner function output on the x - axis of the graph of the outer function.
  4. 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 ) ) .

Explanation of the composite function.

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

what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
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Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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salma
Commplementary angles
Idrissa Reply
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Uday
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salma
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Ayuba
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Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
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Practice Key Terms 1

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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