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