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HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 4.5: Combining Functions
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Hawkes Learning Systems:College Algebra
Section 4.5: Combining Functions
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Objectives
o Combining functions arithmetically.o Composing functions.o Decomposing functions.
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Combining Functions Arithmetically
Addition, Subtraction, Multiplication and Division of Functions1.
2.
3.
4.
The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individually.
f g x f x g x
f g x f x g x
f g x f x g x
prov, ided that 0 f xf x g x
g g x
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Example: Combining Functions Arithmetically
Given that solve:
a.
22 3 4 and 2f x x x g x x
f g x
f x g x
22 3 4 2x x x 22 4x x
Remember that . f g x f x g x
Continued on the next slide…
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.Example: Combining Functions Arithmetically
(cont.)Given that solve:
b.
22 3 4 and 2f x x x g x x
f g x
f x g x
22 3 4 2x x x 3 24 6 8x x x
Remember that . f g x f x g x
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems. All rights reserved.
Example: Combining Functions Arithmetically
Given that find a. and b.
a.
b.
and 2 4 2 3f g
2f g
2fg
2 2f g
4 3
7
22
fg
43
Remember that . f g x f x g x
Remember that .
xf xf
g g x
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Composing Functions
Composing FunctionsLet f and g be two functions. The composition of f and g, denoted , is the function defined by
.
The domain of consists of all x in the domain of g for which g(x) is in turn in the domain of f. The function is read “f composed with g,” or “f of g.”
f g f g x f g x
f g
f g
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Composing Functions
Caution!Note that the order of f and g is important. In general, we can expect the function to be different from the function . In formal terms, the composition of two functions, unlike the sum and product of two functions, is not commutative.
f gg f
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Composing Functions
The diagram below is a schematic of the composition of two functions. The ovals represent sets, with the leftmost oval being the domain of the function g. The arrows indicate the element that x is associated with by the various functions.
x g x f g x
f g
g f
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Example: Composing Functions
Given f(x) = x2 and g(x) = x + 5 , find: a.
= 112
= 121
6f g 6 6 5 11g
6 6f g gf
11f
First, we will find g(6) by replacing x with 6 in g(x).Next, we know that f composed with g can also be written . Since we already evaluated g(6), we can insert the answer to get f(11).
6f g
Continued on the next slide…
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Example: Composing Functions (cont.)
Given f(x) = x2 + 2 and g(x) = x + 5 , find: b.
= (x + 5)2 + 2
= x2 +10x + 25 + 2
= x2 +10x + 27
f g x f g x
5f x
Again, we know by definition that . f g x f g x
Note: since we solved for the variable x we should be able to plug 6 into x and get the same answer as in part a. Verify this.
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Example: Composing Functions
Let f(x) = x – 6 and g(x) = . Simplify the composition and find the domain for:
= g(x – 6)
=
Domain: [6, )
g f g f x
The domain of must be any x such that x – 6 > 0 since x - 6 is under a radical.
g f
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Decomposing Functions
Often functions can be best understood by recognizing them as a composition of two or more simpler functions. For example, the function can be thought of as the composition of two or more functions. Note: if then:
32h x x
3 and 2gf x x x x
2g x ff f xg x
32x
.( )h x
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Decomposing FunctionsEx: The function can be written as a composition of functions in many different ways. Some of the decompositions of f(x) are shown below:a.
b.
c.
23 3 5f x x
3g x x
23 5h x x
3 5g x x
23h x x
3g x x
3 5h x x
2i x x
g h x 23 5g x 23 3 5x f x
23g x 3 2 53x f x
2g h x 23 5g x
g h x
g h i x
23 3 5x f x
HAWKES LEARNING SYSTEMS
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Example: Decomposing Functions
Decompose the function into:a. a composition of two functions
b. a composition of three functions
Note: These are NOT the only possible solutions for the decompositions of f(x)!
4 5 1f x x
4
1
5h x
g x
x
x
4
1
5h
x
x
i
x
g
x
x
x
g h x 4 5g x 4 5 1x f x
g h i x 4g h x 4 5g x 4 5 1x f x
