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5.1
Combining
Functions
♦ Perform arithmetic operations on functions
♦ Perform composition of functions
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Five Ways of Combining Two Functions
f and g
• Addition
•
• Subtraction
•
• Multiplication
•
• Division
•
• Composition
•
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Definition-Addition
If f(x) and g(x) both exist, the sum, of two functions f and g
are defined by
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 3
Example of Addition of Functions:
Let f(x) = x2 + 2x and g(x) = 3x - 1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Definitions-Subtraction
If f(x) and g(x) both exist, the difference of two functions f
and g are defined by
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 5
Example of Subtraction of Functions:
Let f(x) = x2 + 2x and g(x) = 3x  1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Examples of Evaluating
Combinations of Functions –
Using Symbolic Representations
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 7
Definitions-Multiplication
If f(x) and g(x) both exist, the product of two functions f
and g are defined by
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 8
Example of Multiplication of Functions:
Let f(x) = x2 + 2x and g(x) = 3x  1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Definitions-Division
If f(x) and g(x) both exist, quotient of two functions f and g
are defined by
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Example of Division of Functions:
Let f(x) = x2 + 2x and g(x) = 3x  1
• Find the symbolic representation for the
function f and use this to evaluate  f ( 2 )
g
g 
f 
x 2  2( x )
•  x  
3( x )  1
g 
f 
22  2(2) 8
 2 

• So  g 
3(2)  1 5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 11
Definitions-Composition
If f(x) and g(x) both exist, the composition of two functions
f and g are defined by
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Composition of Functions-Symbolic
Find a symbolic representation for the composite function
g ○ f that converts x miles into inches.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 13
Example of Composition of Functions:
Let f(x) = x2 + 2x and g(x) = 3x – 1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 14
Product and Composition of Two
Functions
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Slide 5- 15
Evaluating Combinations of Functions
Numerically
• Given numerical
representations for f
and g in the table
• Evaluate
combinations of f
and g as specified.
x
f(x)
g(x)
x
(f + g)(x)
(f – g)(x)
(fg)(x)
(f/g)(x)
(f  g)(x)
5
8
6
6
7
5
5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7
6
8
6
8
5
7
7
8
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Evaluating Combinations of Functions
Graphically
• Use graph of f and g below to evaluate
• (f + g) (1)
• (f – g) (1)
• (f  g) (1)
• (f/g) (1)
• (f  g) (1)
y
y = f(x)
4
3
2
1
0
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
-3
-4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y = g(x)
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Answers:
y
y = f(x)
4
3
2
1
0
-4
-3
-2
-1
0
1
2
3
4
x
-1
-2
-3
-4
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
y = g(x)
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