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171S2.3p The Composition of Functions
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September 25, 2012
Sep 193:04 PM
CHAPTER 2: More on Functions
MAT 171 Precalculus AlgebraDr. Claude Moore
Cape Fear Community College
This 6minute video covers the Composition of Fucntions. It is available at http://www.khanacademy.org/video/functionspart4?playlist=Algebra
2.1 Increasing, Decreasing, and Piecewise Functions; Applications2.2 The Algebra of Functions2.3 The Composition of Functions2.4 Symmetry2.5 Transformations 2.6 Variation and Applications
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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.
Composition of FunctionsDefinition: The composite function f o g, the composition of f and g, is defined as
(f o g)(x) = f(g(x)), where x is in the domain of g and g(x) is in the domain of f.Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
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Example
Given that f(x) = 3x − 1 and g(x) = x2 + x − 3, find:
a) b)
c)
d)
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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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ExampleIf 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.
Decomposing a Function as a CompositionIn 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.
Sep 209:30 PM
Section 2.3 Composition of Functions
188/4. Given that f(x) = 3x + 1, g(x) = x2 2x 6, and h(x) = x3, find (g o h)(1/2).
171S2.3p The Composition of Functions
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September 25, 2012
Sep 209:30 PM
Section 2.3 Composition of Functions
188/8. Given that f(x) = 3x + 1, g(x) = x2 2x 6, and h(x) = x3, find (h o g)(3).
Sep 209:30 PM
Section 2.3 Composition of Functions
188/10. Given that f(x) = 3x + 1, g(x) = x2 2x 6, and h(x) = x3, find (g o g)(3).
Sep 209:30 PM
Section 2.3 Composition of Functions
188/12. Given that f(x) = 3x + 1, g(x) = x2 2x 6, and h(x) = x3, find (h o h)(1).
Sep 209:30 PM
Section 2.3 Composition of Functions
189/24. Find (f o g)(x) and (g o f)(x) and the domain of
each for the functions: .
Sep 209:30 PM
Section 2.3 Composition of Functions
189/28. Find (f o g)(x) and (g o f)(x) and the domain of each for the functions: f(x) = √x and g(x) = 2 3x.
Sep 209:30 PM
Section 2.3 Composition of Functions
189/34. Find (f o g)(x) and (g o f)(x) and the domain of each for the functions: f(x) = 1 x2 and .
171S2.3p The Composition of Functions
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September 25, 2012
Sep 209:30 PM
Section 2.3 Composition of Functions
189/36. Find (f o g)(x) and (g o f)(x) and the domain of
each for the functions: .
Sep 209:30 PM
Section 2.3 Composition of Functions
189/40. Find f(x) and g(x) such that h(x) = (f o g)(x) given that . Answers may vary.
Sep 209:30 PM
Section 2.3 Composition of Functions
189/42. Find f(x) and g(x) such that h(x) = (f o g)(x)
given that . Answers may vary.
Sep 209:30 PM
Section 2.3 Composition of Functions
189/43. Find f(x) and g(x) such that
h(x) = (f o g)(x) given that . Answers may vary.
Sep 209:30 PM
Section 2.3 Composition of Functions
189/46. Find f(x) and g(x) such that
h(x) = (f o g)(x) given that . Answers may vary.
Sep 232:22 PM
189/51. Ripple Spread. A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3 ft/sec.a) Find a function r(t) for the radius in terms of t. b) Find a function A(r) for the area of the ripple in terms of the radius r. c) Find A(r(t)). Explain the meaning of this function.
171S2.3p The Composition of Functions
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September 25, 2012
Sep 232:22 PM
189/51. Ripple Spread. A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3 ft/sec.a) Find a function r(t) for the radius in terms of t. b) Find a function A(r) for the area of the ripple in terms of the radius r. c) Find A(r(t)). Explain the meaning of this function.
Sep 232:22 PM
189/51. Ripple Spread. A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3 ft/sec.a) Find a function r(t) for the radius in terms of t. b) Find a function A(r) for the area of the ripple in terms of the radius r. c) Find A(r(t)). Explain the meaning of this function.
Sep 232:22 PM
189/51. Ripple Spread. A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 3 ft/sec.a) Find a function r(t) for the radius in terms of t. b) Find a function A(r) for the area of the ripple in terms of the radius r. c) Find A(r(t)). Explain the meaning of this function.