Chapter 7 Functions and Graphs

7.2 Domain and Range Determining the Domain and the Range Restrictions on Domain Piecewise-Defined Functions

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.

Find the domain and range of the function f below. 6 5 4 2 3 -4 1 -2 -1 -3 -5 -4 -3 -2 -1 1 2 3 4 5 f -5 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}.

For the function f represented below, determine each of the following. y x -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -1 -4 -3 3 2 5 1 6 7 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

Solution a) What member of the range is paired with -2 x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -1 -4 -3 3 2 5 1 6 Input Output 7 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.

Solution b) The domain of f x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -1 -4 -3 3 2 5 1 6 The domain of f 7 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

Solution c) What member of the domain is paired with 6 x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -1 -4 -3 3 2 5 1 6 Input Output 7 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” 2.5.24

Solution d) The range of f x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -1 -4 -3 3 2 5 1 6 The range of f 7 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

Determine the domain of Solution We ask, “Is there any number x for which we cannot compute 3x2 – 4?” Since the answer is no, the domain of f is the set of all real numbers.

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. To determine what x-value would cause x – 8 to be 0, we solve an equation: 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

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.

Find each function value for the function f given by a. f(5) b 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

Find each function value for the function f given by a. f(5) b 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

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) = 3 + 3 = 0 b)f(x) = x2; f(2) = 22 = 4 c)f(7) = 4x = 4(7) = 28