171S2.2q The Algebra of Functions
February 13, 2013
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
2.5 Transformations
2.6 Variation and Applications
Mathematica Interactive Figures
are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
Sep 15­9:15 PM
Feb 12­2:03 PM
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.
Sums, Differences, Products, and Quotients of Functions
If f and g are functions and x is in the domain of each function, then Sep 15­9:15 PM
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
Sep 15­9:15 PM
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171S2.2q The Algebra of Functions
February 13, 2013
Another Example
Given that f(x) = x2 + 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.
Remember: f(x) = x2 + 2 and g(x) = x − 3
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.
Mathematica Interactive Figures
are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
Sep 15­9:15 PM
Sep 15­9:15 PM
Another Example
For the function f given by f (x) = x2 + 2x − 3, find the difference quotient.
Example
For the function f given by f (x) = 5x − 1, find the difference quotient
Solution: We first find f (x + h):
Solution: We first find f (x + h):
Sep 15­9:15 PM
Sep 15­9:15 PM
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171S2.2q The Algebra of Functions
February 13, 2013
Section 2.2 Algebra of Functions
180/4. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (fg)(2).
Section 2.2 Algebra of Functions
180/8. Given f(x) = x2 – 3 and g(x) = 2x + 1, find .
180/5. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (f / g)(­1/2).
180/10. Given f(x) = x2 – 3 and g(x) = 2x + 1, find (g / f)(­1/2).
Sep 15­9:28 PM
Sep 15­9:28 PM
Section 2.2 Algebra of Functions
Section 2.2 Algebra of Functions
180/20. 180/24. Sep 15­9:35 PM
Sep 15­9:35 PM
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171S2.2q The Algebra of Functions
Section 2.2 Algebra of Functions
February 13, 2013
Section 2.2 Algebra of Functions
180/32. 180/30. Sep 15­9:38 PM
Sep 15­9:38 PM
Section 2.2 Algebra of Functions
181/36. Find the domain of F – G, FG, and F / G.
Section 2.2 Algebra of Functions
181/36 . Find the domain of F – G, FG, and F / G.
181/38. Graph F + G.
181/40 . Graph F ­ G.
Sep 15­9:39 PM
Sep 15­9:39 PM
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171S2.2q The Algebra of Functions
February 13, 2013
Section 2.2 Algebra of Functions
181/47 . Total Cost, Revenue, and Profit.
In economics, functions that involve revenue, cost, and profit are used. For example, suppose that and 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. 2 ; C(x) = 3x + 13, find each of the Given R(x) = 60x – 0.4x
following:
a) P(x)
Section 2.2 Algebra of Functions
181/47 . Total Cost, Revenue, and Profit.
In economics, functions that involve revenue, cost, and profit are used. For example, suppose that and 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. 2 ; C(x) = 3x + 13, find each of the Given R(x) = 60x – 0.4x
following:
a) P(x)
b) R(100)
b) R(100)
C(100)
C(100)
P(100)
P(100)
Sep 15­9:40 PM
Sep 15­9:40 PM
Section 2.2 Algebra of Functions
Section 2.2 Algebra of Functions
181/48 . Total Cost, Revenue, and Profit.
Given that
R(x) = 200x ­ x 2 and C(x) = 5000 + 8x for a new weather radio produced by Clear Communication, find each of the following. (See Exercise 47.)
a) P(x) b) R(175), C(175), and P(175) c) Using a graphing calculator, graph the three functions in the viewing window [0, 200, 0, 10,000] .
Sep 15­9:40 PM
181/48 . Total Cost, Revenue, and Profit.
Given that
R(x) = 200x ­ x 2 and C(x) = 5000 + 8x for a new weather radio produced by Clear Communication, find each of the following. (See Exercise 47.)
a) P(x) b) R(175), C(175), and P(175) c) Using a graphing calculator, graph the three functions in the viewing window [0, 200, 0, 10,000] .
Sep 15­9:40 PM
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171S2.2q The Algebra of Functions
181/52. For the function f(x) = 5x + 3, construct and simplify the difference quotient
February 13, 2013
181/54. For the function f(x) = (­1/2)x + 7, construct and simplify the difference quotient
Sep 15­9:50 PM
Sep 15­9:50 PM
181/60. For the function f(x) = x2 ­ 3, construct and simplify the difference quotient
181/62. For the function f(x) = 2 ­ x2, construct and simplify the difference quotient
Sep 15­9:50 PM
Sep 15­9:50 PM
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