Download 171S1.2 Functions and Graphs

171S1.2 Functions and Graphs
MAT 171
Dr. Claude Moore, CFCC
CHAPTER 1: Graphs, Functions, and Models
1.1 Introduction to Graphing
1.2 Functions and Graphs
1.3 Linear Functions, Slope, and Applications
1.4 Equations of Lines and Modeling
1.5 Linear Equations, Functions, Zeros and Applications
1.6 Solving Linear Inequalities
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Function
A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.
It is important to note that not every correspondence between two sets is a function.
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1.2 Functions and Graphs
• Determine whether a correspondence or a relation is a function.
• Find function values, or outputs, using a formula or a graph.
• Graph functions.
• Determine whether a graph is that of a function.
• Find the domain and the range of a function.
• Solve applied problems using functions.
Copyright © 2009 Pearson Education, Inc.
Example
Determine whether each of the following correspondences is a function.
This correspondence is a function because each member of the domain corresponds to exactly one member of the range. The definition allows more than one member of the domain to correspond to the same member of the range.
b.
Helen Mirren
Jennifer Hudson
Leonardo DiCaprio
Jamie Foxx
The Queen
Blood Diamond
Dreamgirls
The Departed
This correspondence is not a function because there is one member of the domain (Leonardo DiCaprio) that is paired with more than one member of the range (Blood Diamond and The Departed).
Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
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171S1.2 Functions and Graphs
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Relation
Example
A relation is a correspondence between the first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range.
Determine whether each of the following relations is a function. Identify the domain and range.
a. {(9, ­5), (9, 5), (2, 4)}
Not a function. Ordered pairs (9, –5) and (9, 5) have the same first coordinate and different second coordinates.
Domain is the set of first coordinates: {9, 2}. Range is the set of second coordinates: {–5, 5, 4}.
Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
Example (continued)
Example (continued)
Determine whether each of the following relations is a function. Identify the domain and range.
Determine whether each of the following relations is a function. Identify the domain and range.
b. {(–2, 5), (5, 7), (0, 1), (4, –2)}
c. {(–5, 3), (0, 3), (6, 3)}
Is a function. No two ordered pairs have the same first coordinate and different second coordinates.
Domain is the set of first coordinates: {–2, 5, 0, 4}. Range is the set of second coordinates: {5, 7, 1, –2}.
Copyright © 2009 Pearson Education, Inc.
Is a function. No two ordered pairs have the same first coordinate and different second coordinates.
Domain is the set of first coordinates: {–5, 0, 6}. Range is the set of second coordinates: {3}.
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171S1.2 Functions and Graphs
Notation for Functions
The inputs (members of the domain) are values of x substituted into the equation. The outputs (members of the range) are the resulting values of y.
f (x) is read “f of x,” or “f at x,” or “the value of f at x.”
January 10, 2012
Example
A function is given by f(x) = 2x2 ­ x + 3. Find each of the following.
a. f (0) b. f (–7) c. f (5a) d. f (a – 4)
a. f (0)
f (0) = 2(0)2 − 0 + 3 = 0 – 0 + 3 = 3
b. f (–7)
f (–7) = 2(–7)2 − (–7) + 3 = 2 • 49 + 7 + 3 = 108
Copyright © 2009 Pearson Education, Inc.
Copyright © 2009 Pearson Education, Inc.
Example (continued)
Graphs of Functions
2 A function is given by f(x) = 2x ­ x + 3. Find each of the following.
a. f (0) b. f (–7) c. f (5a) d. f (a – 4)
c. f (5a)
f (5a) = 2(5a)2 − 5a + 3 = 2 • 25a2 – 5a + 3
= 50a2 – 5a + 3
d. f (a – 4)
f (a – 4) = 2(a – 4)2 − (a – 4) + 3
= 2(a2 – 8a + 32) – a + 4 + 3
= 2a2 – 16a + 64 – a + 4 + 3
= 2a2 – 17a + 71
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We graph functions the same way we graph equations. We find ordered pairs (x, y), or (x, f (x)), plot the points and complete the graph.
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171S1.2 Functions and Graphs
Example
January 10, 2012
Example (continued)
Graph f (x) = x3 – x .
2
Graph f (x) = x – 5 .
Make a table of values.
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Graph
Copyright © 2009 Pearson Education, Inc.
Vertical­Line Test
Example
For the function f (x) = x2 – 5, use the graph to find each of the following function values.
a. f (3) b. f (–2)
a. Locate the input 3 on the horizontal axis, move vertically (up) to the graph of the function, then move horizontally (left) to find the output on the vertical axis.
f (3) = 4
If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.
Which of graphs (a) ­ (c) (in red) are graphs of functions?
b. Locate the input –2 on the horizontal axis, move vertically (down) to the graph, then move horizontally (right) to find the output on the vertical axis.
f (–2) = –1
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Yes
No
No
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171S1.2 Functions and Graphs
Finding Domains of Functions
When a function f whose inputs and outputs are real numbers is given by a formula, the domain is understood to be the set of all inputs (x­values) for which the expression is defined as a real number. When an input results in an expression that is not defined as a real number, we say that the function value does not exist and that the number being substituted is not in the domain of the function.
January 10, 2012
Example
Find the indicated function values and determine whether the given values are in the domain of the function.
a. f (1) b. f (3) Since division by 0 is not defined, f (3) does not exist and, 3 is in not in the domain of f.
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Example
Copyright © 2009 Pearson Education, Inc.
Visualizing Domain and Range
Find the domain of the function
Solution: We can substitute any real number in the numerator, but we must avoid inputs that make the denominator 0. Solve x2 + 2x ­ 3 = 0.
(x + 3)(x – 1) = 0
x + 3 = 0 or x – 1 = 0
x = –3 or x = 1
The domain consists of the set of all real numbers except ­3 and 1, or {x|x ≠ ­3 and x ≠ 1}. Copyright © 2009 Pearson Education, Inc.
Keep the following in mind regarding the graph of a function:
Domain = the set of a function’s inputs, found on the horizontal axis;
Range = the set of a function’s outputs, found on the vertical axis.
Copyright © 2009 Pearson Education, Inc.
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171S1.2 Functions and Graphs
Example
Graph the function. Then estimate the domain and range.
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91/9. Determine whether the correspondence is a function.
Correspondence
Domain
Range
A set of people in a town A doctor a person uses A set of doctors
Domain = [–4, ∞)
Range = [0, ∞)
Copyright © 2009 Pearson Education, Inc.
92/26. 92/12. Determine whether the correspondence is a function.
Range
Correspondence
Domain
A set of instruments
An instrument each A set of members person plays
of a rock band 6
171S1.2 Functions and Graphs
93/40. A graph of a function is shown. Using the graph, find the indicated function values; that is, given the inputs, find the outputs.
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94/58. Find the domain of the function. Do not use a graphing calculator.
94/62. Find the domain of the function. Do not use a graphing calculator.
95/80. Graph the function with a graphing calculator. Then visually estimate the domain and the range.
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171S1.2 Functions and Graphs
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