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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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 2 Example Determine whether each of the following correspondences is a function. a. 6 6 3 3 0 36 9 0 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 3 Example Determine whether each of the following correspondences is a function. b. Helen Mirren The Queen Jennifer Hudson Blood Diamond Dreamgirls Leonardo DiCaprio The Departed Jamie Foxx 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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 4 Relation 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 5 Example 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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 6 Example (continued) Determine whether each of the following relations is a function. Identify the domain and range. b. {(–2, 5), (5, 7), (0, 1), (4, –2)} 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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 7 Example (continued) Determine whether each of the following relations is a function. Identify the domain and range. b. {(–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: {–5, 0, 6}. Range is the set of second coordinates: {3}. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 8 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.” Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 9 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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 10 Example (continued) 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) 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 + 16) – a + 4 + 3 = 2a2 – 16a + 32 – a + 4 + 3 = 2a2 – 17a + 39 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 11 Graphs of Functions 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 12 Example Graph f (x) = x2 – 5 . Make a table of values. x f (x) (x, f (x)) 4 3 (3, 4) –1 (2, –1) 2 –1 –4 (–1, –4) 0 5 (0, 5) 1 4 (1, 4) 2 3 1 4 (2, 1) (3, 4) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 13 Example (continued) Graph f (x) = x3 – x . x f (x) (x, f (x)) 3 -24 (3, –24) –6 (2, –6) 2 –1 0 (–1, 0) 0 0 (0, 0) 1 0 (1, 0) 2 3 6 24 (2, 6) (3, 24) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 14 Example (continued) Graph f x x 4 . x 4 f ( x) 0 2 0 2 2 2 6 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 15 Example For the function f (x) = x2 – 6, use the graph to find each of the following function values. a. f (‒3) b. f (1) a. Locate the input ‒3 on the horizontal axis, move vertically (up) to the graph of the function, then move horizontally to find the output on the vertical axis. f (‒3) = 3 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 16 Example For the function f (x) = x2 – 6, use the graph to find each of the following function values. a. f (‒3) b. f (1) b. Locate the input 1 on the horizontal axis, move vertically (down) to the graph, then move horizontally to find the output on the vertical axis. f (1) = –5 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 17 Vertical-Line Test 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 18 Example Which of graphs (a) - (c) (in red) are graphs of functions? Yes. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley No. No. Slide 1.2 - 19 Example (continued) Which of graphs (d) - (f) (in red) are graphs of functions? In graph (f), the solid dot shows that (–1, 1) belongs to the graph. The open circle shows that (–1, –2) does not belong to the graph. No. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Yes. Yes. Slide 1.2 - 20 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 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. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 21 Example Find the indicated function values and determine whether the given values are in the domain of the function. a. f (1) 1 f (x) x3 1 1 f (1) 1 3 2 Since f (1) is defined, 1 is in the domain of f. b. f (3) 1 1 f (3) 3 3 0 Since division by 0 is not defined, f (3) does not exist and, 3 is in not in the domain of f. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 22 Example 3x 2 x 7 . Find the domain of the function h(x) 2 x 2x 3 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 © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 23 Visualizing Domain and Range Keep the following in mind regarding the graph of a function: Domain = the set of a function’s inputs, found on the horizontal axis (x-axis); Range = the set of a function’s outputs, found on the vertical axis (y-axis). Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 24 Example Graph the function. Then estimate the domain and range. f ( x) x 4 Domain = [–4, ) Range = [0, ) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 25 Example Graph the function. Then estimate the domain and range. f ( x) x 4 2 x 2 3 Domain = (–, ) Range = [‒4, ) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Slide 1.2 - 26