1 arithmetic combinations of functions do now: given f (x) = 2x – 3 and g (x) = x 2 – 1. find (f...
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3 Domain for Combination of Functions Combinations of functions: (f + g)(x), (f – g)(x), (fg)(x), (f/g)(x) Any restrictions on the domain of f and the domain of g must also be considered when forming the sum, difference, product, or quotient of f and g. (As well as the combination.)TRANSCRIPT
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Arithmetic Combinations of Functions
Do Now: Given f (x) = 2x – 3 and g (x) = x2 – 1. Find (f + g)(x) and determine the domain.
![Page 2: 1 Arithmetic Combinations of Functions Do Now: Given f (x) = 2x – 3 and g (x) = x 2 – 1. Find (f + g)(x) and determine the domain](https://reader036.vdocuments.us/reader036/viewer/2022081513/5a4d1afc7f8b9ab0599841fb/html5/thumbnails/2.jpg)
Copyright © Cengage Learning. All rights reserved.
Pre-Calculus Honors1.5: Combinations of Functions
HW: p.56-57 (8, 14-18 even, 30, 42, 54, 56, 60)
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Domain for Combination of Functions
Combinations of functions: (f + g)(x), (f – g)(x), (fg)(x), (f/g)(x)Any restrictions on the domain of f and the domain of g must also be considered when forming the sum, difference, product, or quotient of f and g. (As well as the combination.)
![Page 4: 1 Arithmetic Combinations of Functions Do Now: Given f (x) = 2x – 3 and g (x) = x 2 – 1. Find (f + g)(x) and determine the domain](https://reader036.vdocuments.us/reader036/viewer/2022081513/5a4d1afc7f8b9ab0599841fb/html5/thumbnails/4.jpg)
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Compositions of Functions
Another way of combining two functions is to form the composition of one with the other.
For instance, when f (x) = x2 and g (x) = x + 1, the composition of f with g is
f (g (x)) = f (x + 1) = (x + 1)2.
This composition is denoted as f g and is read as “f composed with g”.
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Compositions of Functions
Figure 1.50
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Forming the Composition of f with g
Given and g (x) = x – 1, x 1.Evaluate and determine the domain for part c.a.) (f g)(2)
b.) (g f)(0)
c.) (f g)(x)
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Forming the Composition of f with g
Evaluate (f g)(x), (g f)(x), and determine the domain given
41)(,2)( 2
x
xgxxf
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Do Now: Evaluate the combination of functions.
1.) (f g)(2) 2.) (f + g)(3)
x 0 1 2 3 4
g(x) 4 3 2 1 0
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9Copyright © Cengage Learning. All rights reserved.
Pre-Calculus Honors1.5: Combinations of FunctionsHW: p.57-58 (68, 70, 81, 84, 88)
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p.55, Example 10 – Bacteria CountThe number N of bacteria in a refrigerated petri dish is given by N(T) = 20T 2 – 80T + 500, 2 T 14, where T is the temperature of the petri dish (in degrees Celsius). When the petri dish is removed from refrigeration, the temperature of the petri dish is given by T (t) = 4t + 2, 0 t 3, where t is the time (in hours).a. Find the composition N (T(t)) and interpret its meaning in this context.b. Find the number of bacteria in the petri dish when t = 2 hrs.c. Find the time when the bacteria count reaches 2000.
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Example 10 – Bacteria Count
Solution:a. N (T(t)) = 20(4t + 2)2 – 80(4t + 2) + 500
= 20(16t2 + 16t + 4) – 320t – 160 + 500
= 320t2 + 320t + 80 – 320t – 160 + 500
= 320t2 + 420The composite function N (T(t)) represents the number of bacteria as a function of the amount of time the petri dish has been out of refrigeration.
cont’d
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Example 10 – Solutionb. When t = 2 the number of bacteria is
N = 320(2)2 + 420
= 1280 + 420
= 1700.c. The bacteria count will reach N = 2000 when 320t2 + 420 = 2000. You can solve this equation for t algebraically as follows.
320t2 + 420 = 2000
cont’d