Transcript 2.3 Combinations of Functions

2.3 Combinations of Functions
Introductory
MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
(12th Edition)
Carol A. Marinas, Ph.D.
Copyright © 2009
Combining Functions
If f(x) = 3x + 1 and g(x) = x2 + 5x, find the following:
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f + g)(x) = f(x) + g(x)
(f – g)(x) = f(x) – g(x)
(f + g)(x) = (3x + 1) + (x2 + 5x)
(f – g)(x) = (3x + 1) – (x2 + 5x)
(f + g)(x) = x2 + 8x + 1
(f – g)(x) = – x2 – 2x + 1
(fg)(x) = f(x) • g(x)
f (x) = f (x) for g(x) ≠ 0
g
g (x)
(fg)(x) = f(x) • g(x)
(fg)(x) = (3x + 1) • (x2 + 5x)
(fg)(x) = 3x3 + 16x2 + 5x
Carol A. Marinas, Ph.D.
f (x) = f (x) for g(x) ≠ 0
g
g (x)
f (x) = 3x + 1 for x ≠ 0, –5
g
x2 + 5x
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Composition of Functions
The composition combines two functions by applying one
function to a number and then applying the other function to
the result.
Range of g
Domain of f
Domain of g
g
X
Carol A. Marinas, Ph.D.
Range of f
f
g(x)
f (g(x)) =
(f ₀ g)(x)
f₀g
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Composition of Functions
The function h(x) = (3x + 1)2 is the composition combines two
functions. The first function is 3x + 1 and the second function
is to square the result. So g(x) = 3x + 1 and f(x) = x2.
g
X
f
g(x) =
3x + 1
(f ₀ g)(x)
= f (g(x))
= f (3x + 1)
= (3x + 1)2
h(x) = (f ₀ g)(x) = (3x + 1)2
Carol A. Marinas, Ph.D.
Copyright © 2009
Composition of Functions
With the function h(x) = (3x + 1)2 , find h(5) using the two-step
composition method.
g
5
Carol A. Marinas, Ph.D.
f
g(x)
= 3•5 + 1
= 16
(f ₀ g)(x)
= f (g(x))
= f (16)
= (16)2
= 256= (16)2 = 256
h(5) =h(5)
(f ₀ g)(5)
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Review of Section 2.3
If f(x) = 1 – x and g(x) = 2x2, find the following:
1.
2.
3.
4.
5.
(f + g ) (3)
(g – f) (x)
(fg)(1)
(gf)(x)
g (x) [state the domain of the answer]
f
6. f(3)
7. g(– 2)
8. (g ⁰ f )(3)
9. (g ⁰ f )(x)
10. (f ⁰ f )(x)
Click mouse to check your answers.
Carol A. Marinas, Ph.D.
ANSWERS
1.
2.
3.
4.
5.
16
2x2 + x – 1
0
– 2x3 + 2x2
2 x2 [All Reals except 1]
1–x
6. – 2
7. 8
8. 8
9. 2(1 – x )2
10. x
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Next: Inverse Functions
The compositions of functions will be
used to prove that two functions are
inverses.
Carol A. Marinas, Ph.D.
Copyright © 2009