Warm up

Up to now, we’ve been looking at functions represented by a single equation. In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain. These are called piecewise functions.

Definition: Piecewise Function –a function defined by two or more functions over a specified domain.

There are two things you can do with piecewise functions: 1.Evaluate the piecewise function at a value 2.Graph the piecewise function

Evaluate f(x) when x=0, x=2, x=4

2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Example 2A: Evaluating a Piecewise Function Evaluate each piecewise function for x = –1 and x = 4.

2 x if x ≤ –1 5x if x > –1 g(x) = Example 2B: Evaluating a Piecewise Function Evaluate each piecewise function for x = –1 and x = 3.

12 if x < –3 20 if x ≥ 6 f(x) = Check It Out! Example 2a Evaluate each piecewise function for x = –1 and x = if –3 ≤ x < 6

3x if x < 0 5x – 2 if x ≥ 0 g(x) = Check It Out! Example 2b Evaluate each piecewise function for x = –1 and x = 3.

Graph:

Example 3A: Graphing Piecewise Functions g(x) = 1 4 x + 3 if x < 0 –2x + 3 if x ≥ 0

g(x) = 1 2 Graph each function. x – 3 if 0 ≤ x < 4 x 2 – 3 if x < 0 (x – 4) 2 – 1 if x ≥ 4

f(x) = Graph the function. 4 if x ≤ –1 –2 if x > –1

g(x) = Graph the function. x + 3 if x ≥ 2 –3x if x < 2

Step Functions

Example 4: Sports Application Jennifer is completing a 15.5-mile triathlon. She swims 0.5 mile in 30 minutes, bicycles 12 miles in 1 hour, and runs 3 miles in 30 minutes. Sketch a graph of Jennifer ’ s distance versus time. Then write a piecewise function for the graph.

Step 1 Make a table to organize the data. Use the distance formula to find Jennifer ’ s rate for each leg of the race. Example 4 Continued

Step 1 Make a table to organize the data. Use the distance formula to find Jennifer ’ s rate for each leg of the race. Example 4 Continued

Step 2 Because time is the independent variable, determine the intervals for the function. Example 4 Continued

Step 2 Because time is the independent variable, determine the intervals for the function. Swimming: 0 ≤ t ≤ 0.5 Biking: 0.5 < t ≤ 1.5 Running: 1.5 < t ≤ 2 She swims for half an hour. She bikes for the next hour. She runs the final half hour. Example 4 Continued

Step 3 Graph the function. After 30 minutes, Jennifer has covered 0.5 miles. On the next leg, she reaches a distance of 12 miles after a total of 1.5 hours. Finally she completes the 15.5 miles after 2 hours. Example 4 Continued

Step 3 Graph the function. After 30 minutes, Jennifer has covered 0.5 miles. On the next leg, she reaches a distance of 12 miles after a total of 1.5 hours. Finally she completes the 15.5 miles after 2 hours. Example 4 Continued

Check It Out! Example 4 Shelly earns $8 an hour. She earns $12 an hour for each hour over 40 that she works. Sketch a graph of Shelly ’ s earnings versus the number of hours that she works up to 60 hours. Then write a piecewise function for the graph. Step 1 Make a table to organize the data.

Step 2 Because the number of hours worked is the independent variable, determine the intervals for the function. Check It Out! Example 4 Continued

Step 3 Graph the function. Shelly earns $8 per hour for 0–40. After 40 hours, she earns $12 per hour. Check It Out! Example 4 Continued