Warm-Up Warm-Up Write the equation, domain and range for each graph. Write the equation, domain and range for each graph.

1. 2.

3. f(x) = xf(x) = x22 + 4x - 7, find f(-5). + 4x - 7, find f(-5).

Piecewise FunctionsPiecewise Functions

Objectives: Objectives: Become familiar with piecewise functionsBecome familiar with piecewise functionsEvaluateEvaluate piecewise functions piecewise functions

What Does Research Say?What Does Research Say?

The function concept is one of the central The function concept is one of the central concepts in all of mathematicsconcepts in all of mathematics (Knuth, 2000; (Knuth, 2000; Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, 1993).1993).

Understanding multiple representations of Understanding multiple representations of functions and the ability to move between functions and the ability to move between them is critical to mathematical them is critical to mathematical developmentdevelopment (Knuth, 2000; Rider, 2007).(Knuth, 2000; Rider, 2007).

Piecewise FunctionsPiecewise 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.”

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)

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 Restricting the domain of a functionof a function

Use transformations Use transformations to make a graph of to make a graph of

What is the domain? all real numbers

f(x) = x2 - 3

Looking at only “part” or a “piece” of the function

How could we define How could we define the domain? the domain?

What rule would you write for this function?

(How could we restrict the original function?)

f(x) = x2 - 3 if x ≥ -2

x ≥ -2

f(x) = x2 - 3

Restricting the domain Restricting the domain of a functionof 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 How could we define the domain? domain?

What rule would you write for this function?

€

x < −2

f(x) = –2x – 5

f(x) = –2x – 5 if x < –2

What rule would you write for this piecewise function?

Piecewise FunctionsPiecewise Functions

€

f (x) =

€

−2x − 5 if x < −2

x2 – 3 if x ≥ –2

a) What is the value of y when x = –4? Give two ways to find it.

Piecewise FunctionsPiecewise Functions

€

f (x) =

€

−2x − 5 if x < −2

x2 – 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:Piecing it all together: EvaluatingEvaluating Piecewise Functions Piecewise Functions

Find the interval of the domain Find the interval of the domain that contains the x-valuethat contains the x-value

Then use the rule for that Then use the rule for that interval.interval.

€

f (−3) = ___, f (0) = ___, f (2) = ___, f (7) = ___

€

f (x) =−2x + 3 if x ≤ 2

(x − 2)2 if x > 2

⎧ ⎨ ⎪

⎩ ⎪

9 3 -1 25

2x + 1 if x ≤ 2 x2 – 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) = 42 – 4 = 12

Evaluate the piecewise function for:

x = –1 and x = 4.

3x2 + 1 if x < 0

5x – 2 if x ≥ 0 g(x) =

Because –1 < 0, use the rule for x < 0.Because –1 < 0, use the rule for x < 0.

Because 3 ≥ 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)2 + 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