Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 1 Chapter 7 Functions and Graphs

-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Domain and Range Determining the Domain and the Range Restrictions on Domain Piecewise-Defined Functions 7.2

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 3 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.

-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find the domain and range of the function f below. Here f can be written {(–5, 1), (1, 0), (3, –5), (4, 3)}. The domain is the set of all first coordinates, {–5, 1, 3, 4}. The range is the set of all second coordinates, {1, 0, –5, 3}.

-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example For the function f represented below, determine each of the following. a) What member of the range is paired with -2 b) The domain of f c) What member of the domain is paired with 6 d) The range of f y x f

-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. a) What member of the range is paired with -2 Solution Locate -2 on the horizontal axis (this is where the domain is located). Next, find the point directly above -2 on the graph of f. From that point, look to the corresponding y-coordinate, 3. The “input” -2 has the “output” 3. x y f Input Output 7

-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Solution x y f The domain of f 7 b) The domain of f The domain of f is the set of all x-values that are used in the points on the curve. These extend continuously from −5 to 3 and can be viewed as the curve’s shadow, or projection, on the x-axis. Thus the domain is

-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. c) What member of the domain is paired with 6 Solution Locate 6 on the vertical axis (this is where the range is located). Next, find the point to the right of 6 on the graph of f. From that point, look to the corresponding x- coordinate, 2.5. The “output” 6 has the “input” x y f Input Output 7

-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. d) The range of f Solution The range of f is the set of all y-values that are used in the points on the curve. These extend continuously from -1 to 7 and can be viewed as the curve’s shadow, or projection, on the y-axis. Thus the range is

-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Determine the domain of Solution We ask, “Is there any number x for which we cannot compute 3x 2 – 4?” Since the answer is no, the domain of f is the set of all real numbers.

-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Determine the domain of Solution We ask, “Is there any number x for which cannot be computed?” Since cannot be computed when x – 8 = 0 the answer is yes. x – 8 = 0, x = 8 Thus 8 is not in the domain of f, whereas all other real numbers are. The domain of f is To determine what x-value would cause x – 8 to be 0, we solve an equation:

-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Piecewise-Defined Functions Some functions are defined by different equations for various parts of their domains. Such functions are said to be piecewise-defined. For example, the function given by f(x) = |x| can be described by To evaluate a piecewise-defined function for an input a, we determine what part of the domain a belongs to and use the appropriate formula for that part of the domain.

-13 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find each function value for the function f given by a. f(5)b. f(–8) Solution a. Determine which equation to use. 5 is in the second part of the domain

-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find each function value for the function f given by a. f(5)b. f(–8) Solution b. Determine which equation to use. –8 is in the first part of the domain

-15 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find each function value for the function f given by a) f(  3)b) f(2)c) f(7) Solution a) f(x) = x + 3: f(  3) =  = 0 b)f(x) = x 2 ; f(2) = 2 2 = 4 c)f(7) = 4x = 4(7) = 28