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 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Determine whether each of the following correspondences is a function. a. 6 6 3 3 36 9 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Determine whether each of the following correspondences is a function. 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. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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) 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 + 32) – a + 4 + 3 = 2a2 – 16a + 64 – a + 4 + 3 = 2a2 – 17a + 71 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Graph f (x) = x2 – 5 . Make a table of values. x f (x) (x, f (x)) 3 4 (3, 4) 2 –1 (2, –1) –4 (–1, –4) 5 (0, 5) 1 4 (1, 4) 2 1 (2, 1) 3 (3, 4) Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example (continued) Graph f (x) = x3 – x . Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example (continued) Graph Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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) 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 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Which of graphs (a) - (c) (in red) are graphs of functions? Yes. No. No. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. 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. Yes. Yes. Copyright © 2009 Pearson Education, Inc.

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 © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Find the indicated function values and determine whether the given values are in the domain of the function. a. f (1) Since f (1) is defined, 1 is in the domain of f. 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. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example 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.

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; Range = the set of a function’s outputs, found on the vertical axis. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Graph the function. Then estimate the domain and range. Domain = [–4, ) Range = [0, ) Copyright © 2009 Pearson Education, Inc.