2. CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise
Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry and Transformations
2.5 Variation and Applications
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3. 2.3 The Composition of Functions
Find the composition of two functions and the domain of the composition.
Decompose a function as a composition of two functions.
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4. Composition of Functions
Definition:
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5. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) a)
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6. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) b)
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7. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) a)
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8. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Given that f(x) = 3x  1 and g(x) = x2 + x  3, find:
a) b) b)
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9. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
Given , find the
domain of
Solution:
f (x) is not defined for negative radicands. Since the inputs of are the outputs of g, the domain of consists of all the values in the domain of g for which g(x) is nonnegative.
The domain is
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10. Decomposing a Function as a Composition
In calculus, one needs to recognize how a function can be expressed as the composition of two functions. This can be thought of as “decomposing” the function.
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11. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Example
If h(x) = (3x  1)4, find f(x) and g(x) such that
Solution: The function h(x) raises (3x  1) to the fourth power. Two functions that can be used for the composition are:
f(x) = x4 and g(x) = 3x  1.
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