Warm-Up  Write the equation, domain and range for each graph f(x) = x 2 + 4x - 7, find f(-5). 3. f(x) = x 2 + 4x - 7, find f(-5).

Piecewise Functions Objectives: Become familiar with piecewise functions Evaluate piecewise functions

What Does Research Say?  The function concept is one of the central concepts in all of mathematics (Knuth, 2000; Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, 1993).  Understanding multiple representations of functions and the ability to move between them is critical to mathematical development (Knuth, 2000; Rider, 2007).

Piecewise Functions A piecewise function is a function that is a combination of one or more functions.

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.” The rule for a piecewise function is different for different parts, (or pieces), of the domain (x-values) 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).

Restricting the domain of a function  Use transformations to make a graph of  What is the domain? all real numbers f(x) = x 2 - 3

Looking at only “part” or a “piece” of the function How could we define the domain? How could we define the domain? What rule would you write for this function? (How could we restrict the original function?) f(x) = x if x ≥ -2 x ≥ -2 f(x) = x 2 - 3

Restricting the domain of a function What is the domain? All real numbers What is the equation for this graph? f(x) = –2x – 5

Looking at only “part” or a “piece” of the function How could we define the domain? How could we define the domain? What rule would you write for this function? f(x) = –2x – 5 f(x) = –2x – 5 if x < –2

What rule would you write for this piecewise function? Piecewise Functions x 2 – 3 if x ≥ –2

a) What is the value of y when x = –4? Give two ways to find it. Piecewise Functions x 2 – 3 if x ≥ –2 b) Which equation would you use to find the value of y when x = 2? c) Which equation would you use to find the value of y when x = –2?

Piecing it all together: Evaluating Piecewise Functions  Find the interval of the domain that contains the x-value  Then use the rule for that interval. 9325

2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Because –1 ≤ 2, use the rule for x ≤ 2. Because –1 ≤ 2, use the rule for x ≤ 2. Because 4 > 2, use the rule for x > 2. Because 4 > 2, use the rule for x > 2. h(–1) = 2(–1) + 1 = –1 h(4) = 4 2 – 4 = 12 Evaluate the piecewise function for: x = –1 and x = 4.

3x if x < 0 5x – 2 if x ≥ 0 g(x) = Because –1 < 0, use the rule for x < 0. Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13 g(–1) = 3(–1) = 4 Evaluate each piecewise function for: x = –1 and x = 3

12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6 Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6 f(–1) = 15 Evaluate each piecewise function for: x = –1 and x = 3 15 if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6 Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6