171S2.2 The 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
Sums, Differences, Products, and Quotients of Functions
If f and g are functions and x is in the domain of each function, then September 21, 2010
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.
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 The Algebra of Functions
Another Example
September 21, 2010
Another Example continued
Remember: f(x) = x2 + 2 and g(x) = x − 3
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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.
Difference Quotient
The ratio below is called the difference quotient, or average rate of change.
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). 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 The Algebra of Functions
September 21, 2010
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):
Example continued
Section 2.2 Algebra of Functions
187/4. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (fg)(2).
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171S2.2 The Algebra of Functions
September 21, 2010
Section 2.2 Algebra of Functions
Section 2.2 Algebra of Functions
187/8. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (f / g)(­√3).
187/24. f(x) = √ x and g(x) = √ (2 – x).
Section 2.2 Algebra of Functions
187/32. f(x) = √ (x + 6) and g(x) = 1 / x.
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171S2.2 The Algebra of Functions
September 21, 2010
Section 2.2 Algebra of Functions
Section 2.2 Algebra of Functions
188/34. Find the domain of F – G, FG, and F / G.
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:
a) P(x)
b) R(100)
C(100)
P(100)
188/45 continued
c) Using a graphing calculator, graph the three functions in the viewing window [0, 160, 0, 3000]
188/50 For the function f(x) = 5x + 3, construct and simplify the difference quotient
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