1 of 24 pre-cal chapter 1 sections 5 & 6 sat/act warm - up
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Pre-Cal Chapter 1 Sections 5 & 6
SAT/ACT Warm - up
Chapter 1 Sec 5
Combination Functions
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Pre-Cal Chapter 1 Sections 5 & 6
Chapter 1.5 – Essential Question
How do you combine two functions to form a new
function?
Key Vocabulary: None
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Pre-Cal Chapter 1 Sections 5 & 6
Operations on Functions
Given f(x) = 3x2 – 4 and g(x) = 4x + 5, Find each function.a. (f + g)(x) = f(x) + g(x)
= 3x2 – 4 + 4x + 5= 3x2 + 4x + 1
b. (f - g)(x) = f(x) - g(x) = 3x2 – 4 – (4x + 5)= 3x2 – 4 – 4x – 5= 3x2 – 4x – 9
= (3x2 – 4)(4x + 5)= 12x3 + 15x2 – 16x – 20
c. f g x f x g x
d. f
g
x
f x g x
g x 0
3x 2 44x 5
,
x 5
4
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Pre-Cal Chapter 1 Sections 5 & 6
Composition of Functions
Function can also be combined by using composition. In composition, a function is performed, and then a second function is performed on the result of the first function.
Think in term of manufacturing, a fiber is made into cloth, then cloth is made into garment.
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Pre-Cal Chapter 1 Sections 5 & 6
Composition of FunctionsIn composition, a function g maps the elements in set R to those in set S Another function f maps the elements in set S to those in set T. So the range of g becomes the domain of f.
The function formed by composing two functions f and g is called the composite of f and g , denote by read as “f composition g” or “f of g”
f g,
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Pre-Cal Chapter 1 Sections 5 & 6
Composite Functions
= f(5x – 6) = 2(5x – 6)2 – 3(5x – 6) + 8= 2(25x2 – 60x + 36) – 15x +18 + 8= 50x2 – 135x + 98
= g(2x2 – 3x + 8) = 5(2x2 – 3x + 8) – 6 = 10x2 – 15x + 40 – 6 = 10x2 – 15x + 34
Define Composition of Two Functions
The composition of the function f with the function g is: (f ◦ g)(x) = f(g(x)),The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the
domain of f .
Find (f ◦ g)(x) and (g ◦ f)(x) for f (x) = 2x2 – 3x + 8 and g(x) = 5x – 6.
(f ◦ g)(x) = f(g(x)) (g ◦ f)(x) = g(f(x))
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Pre-Cal Chapter 1 Sections 5 & 6
Composite Functions
What do we know:
1. The domain of f seems to be all real numbers.
2. But the domain of gRemember no negative under radicals is [–3 , 3]
3. So, the domain of (f ◦ g) is [–3 , 3].
Find the domain of the composition (f ◦ g)(x) for the function given by f(x) = x2 – 9 and g(x) = .
(f ◦ g)(x) = f(g(x))
29 x
29 xf
22
22
99
99
xx
x
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Pre-Cal Chapter 1 Sections 5 & 6
Identify a Composite Function
Solution
Write the function h(x) = (3x – 5)3 as a composition of two functions.
One way to write h as a composition of two functions is to take the inner function to be g(x) = 3x – 5 and the outer function to be f(x) = x3. Then you can write
h(x) = (3x – 5)3
= f(3x – 5) = f(g(x)).
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Pre-Cal Chapter 1 Sections 5 & 6
Identify a Composite Function
Solution
Write the function as a composition of two functions.
One way to write h as a composition of two functions is to take the inner function to be g(x) = x – 2 and the outer function to be Then you can write:
22
1
xxh
.1 2
2 x
xxf
22
1
xxh 22 xxh
2 xf xgf
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Pre-Cal Chapter 1 Sections 5 & 6
Chapter 1.5 – Essential Question
How do you combine two functions to form a new
function?
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Pre-Cal Chapter 1 Sections 5 & 6
Daily Assignment• Chapter 1 Sections 5• Text Book (TB)
• Textbook Pgs 58 – 59 #5, 7, 11, 15, 17, 21, 35, 36, 39, 44, 61, 63;
• Read Sections 1.6• Show all work for credit.