171S2.2 Algebra of Functions
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry and Transformations 2.5 Variation and Applications
Go to khanacademy.org, scroll down to Algebra, and look for
Sum and difference; Product and quotient of two functions http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_5/bca05_ifig_launch.html
Sums, Differences, Products, and Quotients of Functions
If f and g are functions and x is in the domain of each function, then February 09, 2012
2.2 The Algebra of Functions
• Find 1. the sum, 2. the difference, 3. the product, 4. the quotient of two functions, 5. determine the domains of the resulting functions.
• Find the difference quotient for a function. See the animation about the algebra of functions.
It is in Course Documents of CourseCompass.
Explanations and Exploratory Exercises for Interactive Figures http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_
5/hidden/bca05_explorations.html
Interactive Figures from Pearson Education: Sum and Difference of Two Functions
Product and Quotient of Two Functions
http://media.pearsoncmg.com/aw/aw_bittinger_colalgebra_5/bca05_ifig_launch.html
Example
Given that f (x) = x + 2 and g(x) = 2x + 5, find each of the following.
a) (f + g)(x)
b) (f + g)(5)
Solution: a)
b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true.
f(5) = 5 + 2 = 7 g(5) = 2(5) + 5 = 15
(f + g)(5) = f(5) + g(5) = 7 + 15 = 22 or using the function from (a) above (f + g)(x) = 3x + 7, we get
(f + g)(5) = 3(5) + 7 = 22
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171S2.2 Algebra of Functions
Another Example
Given that f(x) = x + 2 and g(x) = x − 3, find each of the following.
a) The domain of f + g, f − g, fg, and f/g
b) (f − g)(x)
c) (f/g)(x)
Solution: a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g, f − g, and fg are the set of numbers in the intersection of the domains—that is, the set of numbers in both domains, or all real numbers.
For f/g, we must exclude 3, since g(3) = 0.
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Another Example continued
Remember: f(x) = x2 + 2 and g(x) = x − 3
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b) (f − g)(x) = f(x) − g(x) = (x2 + 2) − (x − 3)
= x2 − x + 5
c) (f/g)(x) =
Remember to add the stipulation that x ≠ 3, since 3 is not in the domain of (f/g)(x). Difference Quotient
The ratio below is called the difference quotient, or average rate of change.
Example
For the function f given by f (x) = 5x − 1, find the difference quotient
Solution: We first find f (x + h):
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171S2.2 Algebra of Functions
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Example continued
Another Example
For the function f given by f (x) = x2 + 2x − 3, find the difference quotient.
Solution: We first find f (x + h):
Section 2.2 Algebra of Functions
187/4. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (fg)(2).
Example continued
﴾f g﴿﴾2﴿ = f ﴾2﴿ ・ g ﴾2﴿
= ﴾2 − 3﴿﴾2 ・ 2 + 1﴿
= 1・ 5 = 5
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171S2.2 Algebra of Functions
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Section 2.2 Algebra of Functions
187/24. Section 2.2 Algebra of Functions
187/8. Given f(x) = x2 – 3 and g(x) = 2x + 1, find .
Solution:
Section 2.2 Algebra of Functions
187/32. 4
171S2.2 Algebra of Functions
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187/32. Section 2.2 Algebra of Functions
188/34. Find the domain of F – G, FG, and F / G.
Section 2.2 Algebra of Functions
188/45. Total Cost, Revenue, and Profit. In economics, functions that involve revenue, cost, and profit are used. For example, suppose that R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new kind of tool for King Hardware Wholesalers. Then the difference P(x) = R(x) – C(x) represents the total profit for producing x tools. Given R(x) = 60x – 0.4x2; C(x) = 3x + 13, find each of the following:
188/45 continued
c) Using a graphing calculator, graph the three functions in the viewing window [0, 160, 0, 3000]
a) P(x)
b) R(100)
C(100)
P(100)
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171S2.2 Algebra of Functions
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188/50 For the function f(x) = 5x + 3, construct and simplify the difference quotient
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