Functions and Their Graphs

2 Identify and graph linear and squaring functions. Recognize EVEN and ODD functions Identify and graph cubic, square root, and reciprocal function. Identify and graph step and other piecewise-defined functions. Recognize graphs of parent functions. OBJECTIVE :

3 Linear and Squaring Functions The graph of the linear function y= mx+b The domain of the function is the set of all real numbers. The range of the function is the set of all real numbers. The graph has an x-intercept of (–b/m, 0) and a y-intercept of (0, b). The graph is increasing if slope m  0, and decreasing if slope m  0, and constant (horizontal line) if m = 0.

4 Example 1 – Writing a Linear Function Write the linear function for which f (1) = 3 and f (4) = 0. Solution: To find the equation of the line that passes through (x 1, y 1 ) = (1, 3) and (x 2, y 2 ) = (4, 0) first find the slope of the line.

5 Example 1 – Solution Next, use the point-slope form of the equation of a line. y – y 1 = m(x – x 1 ) y – 3 = –1(x – 1) y = –x + 4 f (x) = –x + 4 cont’d Point-slope form Substitute for x 1, y 1 and m Simplify. Function notation

6 Example 1 – Solution The graph of this function is shown in Figure cont’d Figure 1.65

7 Linear and Squaring Functions There are two special types of linear functions, the constant function and the identity function. A constant function has the form f (x) = c and has the domain of all real numbers with a range consisting of a single real number c. The graph of a constant function is a horizontal line, as shown in Figure The identity function has the form f (x) = x. Figure 1.66

8 Linear and Squaring Functions Its domain and range are the set of all real numbers. The identity function has a slope of m = 1 and a y-intercept at (0, 0). The graph of the identity function is a line for which each x-coordinate equals the corresponding y-coordinate. The graph is always increasing, as shown in Figure Figure 1.67

9 Linear and Squaring Functions The graph of the squaring function f (x) = x 2 is a U-shaped curve with the following characteristics. The domain of the function - all real numbers. The range - all nonnegative real numbers. The function is even : f(-x) = f(x) The graph has an intercept at (0, 0).

10 Linear and Squaring Functions The graph is decreasing on the interval (, 0) and increasing on the interval (0, ) The graph is symmetric with respect to the y-axis. The graph has a relative minimum at (0, 0). Figure 1.68

11 Cubic, Square Root, and Reciprocal Functions 1. Cubic function f (x) = x 3 The domain - all real numbers. The range - all real numbers. The function is odd : f(-x) = - f(x) f(-x) = (-x) 3 = - x 3 = - f(x)

12 The graph of the cubic function The graph is increasing on the interval (, ). The graph is symmetric with respect to the origin. Cubic function Figure 1.69

13 Square Root Function 2. The graph of the square root function f (x) = The domain – all nonnegative real numbers. The range - all nonnegative real numbers. The graph has an intercept at (0, 0). The Graph is increasing on the interval (0, )

14 The Graph of Square Root Function Square root function Figure 1.70

15 Reciprocal Function 3. The graph of the reciprocal function f (x) = has the following characteristics. The domain of the function is(, 0)  (0, ) The range of the function is (, 0)  (0, ) The function is odd : f(-x) = - f(x) The graph does not have any y or x intercepts. The graph is decreasing on the intervals (, 0) and (0, ). The graph is symmetric with respect to the origin.

16 The graph of the Reciprocal Functions. Reciprocal function Figure 1.71

17 Step and Piecewise-Defined Functions Step functions: Functions whose graphs resemble sets of stair steps The most famous of the step functions is the greatest integer function, which is denoted by and defined as f (x) = = the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. = (greatest integer  –1) = –1 = (greatest integer  ) = –1

18 Step and Piecewise-Defined Functions = (greatest integer  ) = 0 = (greatest integer  1.5) = 1 The graph of the greatest integer function f (x) = The domain of the STEP function - all real numbers. The range of the STEP function - all integers. The graph has a y-intercept at (0, 0) and x-intercepts in the interval [0, 1).

19 Step Function GRAPH Figure 1.72

20 Example 2 – Evaluating a Step Function Evaluate the function when x = –1, 2 and f (x) = + 1 Solution: For x = –1, the greatest integer  –1 is –1, so f (–1) = + 1 = –1 + 1 = 0

21 Example 2 – Solution For x = 2, the greatest integer  2 is 2, so f (2) = + 1 = = 3. For x =, the greatest integer  is 1, so = = 2 cont’d

22 Example 2 – Solution You can verify your answers by examining the graph of f (x) = + 1 shown in Figure cont’d Figure 1.73

23 Parent Functions The most commonly used functions in algebra. (a) Constant Function(b) Identity Function (c) Absolute Value Function (d) Square Root Function Figure 1.75

24 Parent Functions (e) Quadratic Function (f) Cubic Function (g) Reciprocal Function (h) Greatest Integer Function Figure 1.75