Holt Algebra 2

9-2 Piecewise Functions

A piecewise function is a function that is a combination of one or more functions. Piecewise functions are used to evaluate situations where varying rules apply. For instance, movie ticket prices often vary according to a person’s age.

𝑃𝑟𝑖𝑐𝑒 =

$5.50, 𝑎𝑔𝑒𝑠 2 − 12

$8.00, 𝑎𝑔𝑒𝑠 13 − 64

$6.50 𝑎𝑔𝑒𝑠 65 +

Holt Algebra 2

9-2 Piecewise Functions

Each portion of a piecewise function has two

parts: the function and the domain where

that function applies.

𝑓 𝑥 = −𝑥, −∞ < 𝑥 ≤ −1

𝑥2, −1 < 𝑥 < 11, 1 ≤ 𝑥 < ∞

function domain

Holt Algebra 2

9-2 Piecewise Functions

Not all piecewise functions will connect.

There may be gaps or holes in the graph.

𝑓 𝑥 = 1, 𝑥 ≠ 32.5, 𝑥 = 3

Holt Algebra 2

9-2 Piecewise Functions

Be very mindful of open circles (< or >) and

filled-in dots (≤ or ≥).

𝑔 𝑥 =

−2, −∞ < 𝑥 ≤ −33, −3 < 𝑥 < −1

−1, −1 < 𝑥 < 2 3, 2 ≤ 𝑥 < ∞

Holt Algebra 2

9-2 Piecewise Functions

Create a table and function to represent the graph.

Example 1: Consumer Application

Step 1 Create a table

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.

Holt Algebra 2

9-2 Piecewise Functions

The domain of the function is

divided into three intervals:

Weights under 2 [0, 2)

[2, 5)

[5, ∞)

Weights 2 and under 5

Weights 5 and over

Example 1 Continued

Holt Algebra 2

9-2 Piecewise Functions

Example 1 Continued

Step 2 Write an equation.

𝑃 𝑥 = $8, 0 ≤ 𝑥 < 2$6, 2 ≤ 𝑥 < 5$5, 5 ≤ 𝑥 < ∞

Holt Algebra 2

9-2 Piecewise Functions

Example 3A: Graphing Piecewise Functions

g(x) =

1

4

Graph each function.

x + 3 if x < 0

–2x + 3 if x ≥ 0

The function is composed of two linear pieces that will be represented by two rays. Because the domain is divided by x = 0, evaluate both branches of the function at x = 0.

Holt Algebra 2

9-2 Piecewise Functions

Example 3A Continued

For the first branch, the function is 3 when x = 0, so plot the point (0, 3) with an open circle and draw a ray with the slope 0.25 to the left. For the second branch, the function is 3 when x = 0, so plot the point (0, 3) with a solid dot and draw a ray with the slope of –2 to the right.

O ●

Holt Algebra 2

9-2 Piecewise Functions

g(x) =

Graph the function.

x + 3 if x ≥ 2

–3x if x < 2

The function is composed of two linear pieces. The domain is divided at x = 2.

Check It Out! Example 3b

Holt Algebra 2

9-2 Piecewise Functions

Add an open circle at (2, –6) and a closed circle at (2, 5) and so that the graph clearly shows the function value when x = 2.

Check It Out! Example 3b Continued

x –3x x + 3

–4 12

–2 6

0 0

2 –6 5

4 7 O

●

Holt Algebra 2

9-2 Piecewise Functions

2x + 1 if x ≤ 2

x2 – 4 if x > 2

h(x) =

Because –1 ≤ 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 each piecewise function for x = –1 and x = 4.

To evaluate any piecewise function at a specific value, substitute the value into the portion of the function that contains that point.

Holt Algebra 2

9-2 Piecewise Functions

2x if x ≤ –1

5x if x > –1 g(x) =

Because –1 ≤ –1, use

the rule for x ≤ –1.

Because 4 > –1, use the

rule for x > –1. g(4) = 5(4) = 20

g(–1) = 2(–1)

= 1

2

Evaluate each piecewise function for x = –1 and x = 4.

Holt Algebra 2

9-2 Piecewise Functions

Evaluate the following values:

f(-3) =

f(-1) =

f(0) =

f(2) =

1

-2

-3

1

Holt Algebra 2

9-2 Piecewise Functions

Evaluate the following values:

f(-3) =

f(0) =

f(2) =

f(4) =

f(1.999) =

3

1

DNE

1

1.999

Holt Algebra 2

9-2 Piecewise Functions

HW Piecewise Worksheets #1 and #2