Section 1.8
description
Transcript of Section 1.8
Section 1.8Combinations of Functions:
Composite Functions
1st Day
Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows.
1. Sum: (f + g)(x) = f (x) + g(x)
2. Difference: (f − g)(x) = f (x) − g(x)
3. Product: (fg)(x) = f (x) ∙ g(x)
4. Quotient: , 0
f xfx g x
g g x
Example 1
Find (a) (f + g)(x), (b) (f – g)(x), (c) (fg)(x), and
(d) What is the domain of .fx
g
?f
g
1) f (x) = 2x + 5 and g(x) = 2 – x
a) (f + g)(x) = f (x) + g(x)
= (2x + 5) + (2 – x)
= (2x – x) + (5 + 2)
= x + 7
b) (f − g)(x) = f (x) − g(x)
= (2x + 5) − (2 – x)
= (2x + x) + (5 − 2)
= 3x + 3
1) f (x) = 2x + 5 and g(x) = 2 – x
c) (f g)(x) = f (x) ∙ g(x)
= (2x + 5)(2 – x)
= 4x – 2x2 + 10 − 5x
= −2x2 − x + 10
d) , 0
f xfx g x
g g x
2 5
, 22
xx
x
2) f (x) = 3x – 2 and g(x) = x + 7
a) (f + g)(x) = f (x) + g(x)
= (3x − 2) + (x + 7)
= (3x + x) + (−2 + 7)
= 4x + 5
b) (f − g)(x) = f (x) − g(x)
= (3x − 2) − (x + 7)
= (3x − x) + (−2 − 7)
= 2x − 9
2) f (x) = 3x − 2 and g(x) = x + 7
c) (f g)(x) = f (x) ∙ g(x)
= (3x − 2)(x + 7)
= 3x2 + 21x − 2x − 14
= 3x2 + 19x − 14
d) , 0
f xfx g x
g g x
3 2
, 77
xx
x
2
22
3) 4 and 1
xf x x g x
x
a) (f + g)(x) = f (x) + g(x)
22
24
1
xx
x
b) (f − g)(x) = f (x) − g(x)
22
24
1
xx
x
2
22
3) 4 and 1
xf x x g x
x
c) (fg)(x) = f (x) ∙ g(x)
2
22
41
xx
x
2 2
2
4
1
x x
x
22
24
1
xx
x
d) , 0
f xfx g x
g g x
22
2
14
xx
x
2 2
2
1 4, 0
x xx
x
Example 2Evaluate the indicated function for f (x) = x2 +1 and g(x) = x – 4.
1) (f – g)(−1)
(f – g)(−1) = f (−1) − g(−1)
= [(−1)2 + 1] − [−1 − 4]
= 2 − (−5)
= 7
2) (fg)(5) + f (4)
(fg)(5) + f (4) = f (5) ∙ g(5) + f (4)
= [(5)2 + 1][5 − 4] + [(4)2 + 1]
= (26)(1) + 17
= 26 + 17
= 43
3) (f + g)(t – 2)
(f + g)(t – 2) = f (t − 2) + g(t − 2)
= [(t − 2)2 + 1] + [(t − 2) − 4]
= [t2 − 4t + 4 + 1] + [t − 6]
= t2 – 4t + 5 + t – 6
= t2 – 3t – 1
HW: p. 89 (5-23 odd)
2nd DayThe composition of the function f with the function g is:
(f ◦ g)(x) = f (g(x))
The domain of the composition is the set of all x in the domain of g.
Example 1
Given f (x) = x + 2 and g(x) = 4 – x2. Find:
a) (f ◦ g)(x)
b) (g ◦ f )(x)
c) (g ◦ f )(−2)
Given f (x) = x + 2 and g(x) = 4 – x2.
a) (f ◦ g)(x)
(f ◦ g)(x) = f (g(x))
= f (4 – x2)
= [4 – x2] + 2
= 6 – x2
1) Given f (x) = x + 2 and g(x) = 4 – x2.
b) (g ◦ f )(x)
(g ◦ f ) (x) = g (f (x))
= g(x + 2)
= 4 – (x + 2)2
= 4 – [x2 + 4x + 4]
= 4 – x2 – 4x – 4
= –x2 – 4x
c) (g ◦ f )(−2) = –(−2)2 – 4(−2)
= 4
Example 2
Given f (x) = 3x + 5 and g(x) = 5 – x. Find:
a) (f ◦ g)(x)
b) (g ◦ f )(x)
c) (f ◦ f )(x)
Given f (x) = 3x + 5 and g(x) = 5 – x.
a) (f ◦ g)(x)
(f ◦ g)(x) = f (g(x))
= f (5 – x)
= 3[5 – x] + 5
= 15 – 3x + 5
= 20 – 3x
Given f (x) = 3x + 5 and g(x) = 5 – x.
b) (g ◦ f )(x)
(g ◦ f ) (x) = g (f (x))
= g(3x + 5)
= 5 – (3x + 5)
= 4 – 3x – 5
=–3x – 1
Given f (x) = 3x + 5 and g(x) = 5 – x.
c) (f ◦ f )(x)
(f ◦ f ) (x) = f (f (x))
= f (3x + 5)
= 3(3x + 5) + 5
= 9x + 15 + 5
= 9x + 20
Example 3
Given: f (x) = x2 – 9 and
find the composition (f ◦ g)(x). Then find the domain of (f ◦ g)(x).
29 ,g x x
Given: f (x) = x2 – 9 and
(f ◦ g)(x) = f (g(x))
The domain of (f ◦ g)(x) is the domain of g(x),which is [−3, 3].
29 .g x x
29f x
229 9x
29 9x 2x
-4 -3 -2 -1 1 2 3 4
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
Example 4
Given: and g(x) = x + 1, find
(a) (f ◦ g)(x)
(b) (g ◦ f )(x) Find the domain of each function and each composite function.
2
3
1f x
x
Domain of f (x):
Domain of g(x):
, 1 1, 1 1,
,
Given: and g(x) = x + 1, find
(a) (f ◦ g)(x)
(f ◦ g)(x) = f (g(x))
= f (x + 1)
2
3
1f x
x
2
3
1 1x
2
3
2 1 1x x
2
3
2x x
, 2 2, 0 0, Domain of (f ◦ g)(x) :
Given: and g(x) = x + 1, find
(b) (g ◦ f )(x)
(g ◦ f )(x) = g( f (x))
Domain:
2
3
1f x
x
2
3
1g
x
2
31
1x
, 1 1, 1 1,
HW: pp. 89-90 (31-41 odd)