copyright © 2015, 2008, 2011 pearson education, inc. section 5.1, slide 1 chapter 5 logarithmic...
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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 1
Chapter 5Logarithmic Functions
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 3
Composite Function
Definition
If f and g are functions, x is in the domain of g, and g(x) is in the domain of f, then we can form the composite function f ◦ g:
(f ◦ g)(x) = f(g(x))
We say f ◦ g is the composition of f and g.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 4
Example: Using Tables to Evaluate a Composite Function
All of the input-output pairs of functions g and f are shown in the tables below.
1. Find (f ◦ g)(0).2. Use a table to describe five input-output pairs of f ◦ g.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 5
Solution
1. ( )( ) ( )0 ( )0f g f g)8(f
21
2. We repeat the process used in Problem 1 for the inputs 1, 2, 3, and 4 and how the work in the table.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 6
Example: Evaluating a Composite Function
Let f(x) = 2x – 5 and g(x) = 3x + 6.
1. Find (f ◦ g)(4). 2. Find (g ◦ f)(4).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 7
Solution
1. 2.( )( ) ( )4 ( )4f g f g
)8(1f
2( 518)
31
( )( ) ( )4 ( )4g f g f
)3(g
3( 63) 15
Let f(x) = 2x – 5 and g(x) = 3x + 6.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 8
Composite Functions
In general, the outputs (f ◦ g)(a) and (g ◦ f)(a) may or may not be equal.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 9
Example: Finding Equations of Composition Functions
Let f(x) = 2x and g(x) = x + 4.
1. Find an equation of f ◦ g.
2. Find an equation of g ◦ f.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 10
Solution
1. Since g(x) = x + 4, we can substitute x + 4 for g(x) in the second step:
( )( ) ( ( ))f g x f g x
4( )f x 42x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 11
Solution
1. We verify our work by creating a graphing calculator table for y = f(g(x)) and y = 2x + 4.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 12
Solution
2. Since f(x) = 2x, we can substitute 2x for f(x) in the second step.
( )( ) ( ( ))g f x g f x
)2( xg
42x
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 13
Example: Evaluating and Finding an Equation of a Composite Function
Let f(x) = –3x + 8 and g(x) = 4x – 5.
1. Find (f ◦ g)(2).2. Find an equation of f ◦ g.3. Use the equation of f ◦ g to find (f ◦ g)(2).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 14
Solution
1. ( )(2) ( (2))f g f g
)3(f
3( ) 83
1
Let f(x) = –3x + 8 and g(x) = 4x – 5.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 15
Solution
2. Since g(x) = 4x – 5, we can substitute 4x – 5 for g(x) in the second step.
( )( ) ( ( ))f g x f g x5( )4f x 53( 84 )x
12 15 8x 12 23x
Let f(x) = –3x + 8 and g(x) = 4x – 5.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 16
Solution
3.
So, (f ◦ g)(2) is equal to –1, which is the same result we found in Problem 1.
( )( ) ( 32 ) 2212f g 1
Let f(x) = –3x + 8 and g(x) = 4x – 5.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 17
Example: Expressing a Function as a Composition of Two Functions
If h(x) = (7x + 3)5, find equations of the functions f and g such that h(x) = (f ◦ g)(x).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 18
Solution
To form f(g(x)), we substitute g(x) for x in f(x). Similarly, to form (7x + 3)5, we can substitute 7x + 3 for x in x5. This suggests that g(x) = 7x + 3 and f(x) = x5. We check by performing the composition:
(f ◦ g)(x) = f(g(x)) = f(7x + 3) = (7x + 3)5
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 19
Solution
There are other possible answers. For example, g(x) = 7x and f(x) = (x + 3)5 also work:
(f ◦ g)(x) = f(g(x)) = f(7x) = (7x + 3)5
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 20
Example: Using Graphs to Evaluate a Composite Function
Refer to the graph to find (f ◦ g)(5).
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 21
Solution
The blue arrows show that g(5) = –3. So, we can substitute –3 for g(5) in the second step:
( )(5) ( (5))f g f g
3( )f
2
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 22
Example: Using a Composite Function to Model a Situation
Let f(Q) be the number of cups in Q quarts, and let g(x) be the number of ounces in x cups.
1. Find an equation of f.2. Find an equation of g.3. Find an equation of g ◦ f.4. Find (g ◦ f)(5). What does it mean in this situation?
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 23
Solution
1. Since there are 4 cups in one quart, f(Q) = 4Q.
2. Since there are 8 ounces in one cup, g(x) = 8x.
3. (g ◦ f)(Q) = (g(f(Q)) = g(4Q) = 8(4Q) = 32Q
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.1, Slide 24
Solution4. (g ◦ f)(5) = 32(5) = 160
The function f converts units of quarts to units of cups, and the function g converts units of cups to units of ounces, so g ◦ f converts units of quarts to units of ounces.
So, (g ◦ f)(5) = 160 means there are 160 ounces in 5 quarts.
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