Applications of
Piecewise Functions
By Dr. Ma. Louise De Las Peñas
A
piecewise-defined function is a function that is defined using several expressions for different
parts of the domain.
Illustration: Consider the following piecewise function defined by
«3 if x b 1
®
f ( x) ¬1 if 1 x b 2
®4 if x 2
­
We create the graph of f in three parts, as shown and described below.
First, we graph f(x) = -3, only for x values less than or equal to -1. Next, we
graph f(x) = 1, only for x values greater than -1 and less than of equal to 2. Then
we graph f(x) = 4, only for x values greater than 2.
In the next examples, we give applications of piecewise functions used as
mathematical models.
Example 1: Admission fees. A local zoo charges admission to groups according
to the following policy. Groups of fewer than 50 people are charged a rate of
35.00 per person, while groups of 50 people or more are charged a reduced
rate of 30.00 per person.
(a) Find a mathematical model expressing the amount a group will be charged
for admission as a function of its size.
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(b) Sketch the graph of the function in part a)
(c) How much money will a group of 49 people save in admission cost if it can recruit one additional
member?
Solution:
a) Let C(x) be the total admission cost charged to a group, where x is the number of persons in the
group. Then C(x) is a piecewise function defined as follows:
«35 x if 0 b x 50
C ( x) ¬
­30 x if x r 50
b) The graph of function C appears below.
c) C(x) is defined from the equation C(x) = 35x when 0 ≤ x < 50 and from the equation C(x) = 35x when
x ≥ 50. Thus,
C(49) = 35(49) = 1715 while C(50) = 30(50) = 1 500.
Now, 1 715 – 1 500 = 215. Hence a group of 49 people will be able to save
cost if it can recruit one additional member.
215.00 in admission
Example 2: Income tax rates. A certain country taxes the first $20 000 of an individual’s income at a
rate of 15% and all income over $20 000 is taxed at 20%.
(a) Find a piecewise-defined function T that gives the total tax on an income of x dollars.
(b) Sketch the graph of the function in part (a).
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Solution:
(a) Let T(x) be the total tax on an income of x dollars in a certain country. Then T(x) is a piecewise
function defined as follows:
if x b 20, 000
«0.15 x
T ( x) ¬
­0.20 x 1, 000 if x 20, 000
Note that if an individual earns an income x, where x > 20 000, then a tax of 20% is applied for income
beyond 20 000 in addition to a rate of 15% applied to the first 20 000 earned. Thus the formula for tax is
0.20(x-20 000) + 0.15(20 000) that gives 0.20x – 4 000 + 3 000 or 0.20x – 1 000.
(b) The graph of T is given below, which are two intersecting lines.
Example 3: Parking Costs. A parking garage charges 20.00 for up to
(but not including) 1 hour of parking, 40.00 for up to 2 hours parking,
60.00 for up to 3 hours parking and so on.
(a) Find a piecewise-defined function C that gives the cost of parking
for t hours.
(b) Sketch the graph of the function in part (a).
(c) How much will a person pay if he has parked his car for
4.5 hours?
Solution:
(a) The cost function C described here as a function of time t (hours)
is a greatest integer function, defined as follows:
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C(t) = 20||t|| + 20, t > 0
or
C(t) = 20(||t|| + 1), t > 0
For instance, if 0 < t < 1, that is, time has not reached an hour, then the cost is C(t) = 20||t|| + 20 =
20(0) + 20 = 20 pesos.
If 1 ≤ t ≤ 2, then the cost is C(t) = 20||t|| + 20 = 20(1) + 20 = 40 pesos.
If 2 ≤ t ≤ 3, then the cost is C(t) = 20||t|| + 20 = 20(2) + 20 = 60 pesos, and so on.
(b) The graph of C is given below:
(c) If a person has parked his car for 4.5 hours, then his total parking cost will be C(4.5) = 20||4.5|| + 20
= 20(4) + 20 = 100 pesos.
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