Objectives Vocabulary Write and graph piecewise functions. Use piecewise functions to describe real-world situations. Vocabulary piecewise function step function

A piecewise function is a function that is a combination of one or more functions. The rule for a piecewise function is different for different parts, or pieces, of the domain. For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).

Example 1 The domain of the function is divided into three intervals: Weights under 2 0 < x < 1.99 Weights 2 and under 5 2 < x < 4.99 Weights 5 and over x > 5

Green Fee ($) Time Range (h) Example 2 The domain of the function is divided into three intervals: Green Fee ($) Time Range (h) 28 8 A.M. – noon 24 noon – 4 P.M. 12 4 P.M. – 9 P.M. $28 8 < x < 12 $24 12 < x < 4 $12 4 < x < 9

Example 2 Because the endpoints of each segment of the graph identify the intervals of the domain, use the endpoints and points close to them as the domain values in the table.

A piecewise function that is constant for each interval of its domain, such as the ticket price function, is called a step function. You can describe piecewise functions with a function rule. The rule for the movie ticket prices from Example 1 on page 662 is shown.

Read this as “f of x is 5 if x is greater than 0 and less than 13, 9 if x is greater than or equal to 13 and less than 55, and 6.5 if x is greater than or equal to 55.”

To evaluate any piecewise function for a specific input, find the interval of the domain that contains that input and then use the rule for that interval.

Example 3: Evaluating a Piecewise Function Evaluate each piecewise function for x = –1 and x = 4. 2x + 1 if x ≤ 2 h(x) = x2 – 4 if x > 2 h(–1) = h(4) =

Example 4: Evaluating a Piecewise Function 2x if x ≤ –1 g(x) = 5x if x > –1 g(–1) = g(4) =

Example 5 Evaluate each piecewise function for x = –1 and x = 3. 12 if x < –3 f(x) = 15 if –3 ≤ x < 6 20 if x ≥ 6 f(–1) = f(3) =

Example 6 Evaluate each piecewise function for x = –1 and x = 3. 3x2 + 1 if x < 0 g(x) = 5x – 2 if x ≥ 0 g(–1) = g(3) =

You can graph a piecewise function by graphing each piece of the function.

Example 7: Graphing Piecewise Functions Graph each function. 1 4 x + 3 if x < 0 –2x + 3 if x ≥ 0 g(x) =

Example 8: Graphing Piecewise Functions Graph each function. 1 2 g(x)= x – 3 if 0 ≤ x < 4 x2 – 3 if x < 0 (x – 4)2 – 1 if x ≥ 4

Example 9 Graph the function. 4 if x ≤ –1 f(x) = –2 if x > –1 .

Example 10 Graph the function. –3x if x < 2 g(x) = x + 3 if x ≥ 2