Types of Piecewise Functions Part I
Name_________________________
To begin, it is important to remember the difference between a closed dot and an open dot on a
graph.
A closed dot is used to show that a point is a solution of a function or an inequality. For
example, if one graphs x  3 on a number line, a closed dot would be placed at 3 to represent that
it is a solution to the inequality. Graph the inequality on the number line below.
An open dot is used to show that a specific point is NOT a solution to a function or inequality.
For example, if one graphs x  3 on a number, an open dot would be placed at 3 to show that it is
not a solution to the inequality. However, all numbers above 3 are solutions, including 3.01,
3.001, 3.0000000001, and so on. Graph the inequality on the number line below.
--------------------------------------------------------------------------------------------------------------------Graph the piecewise function on the coordinate plane to the right - you will need to use open dots
and solid dots as necessary.
1, 0  x  2

f ( x)  2, 2  x  4
3, 4  x  6

Hint - try to evaluate the function for the following
values of x:
x=0
x=1
x = 0.1
x = 1.5
x = 0.2
x = 1.9999
x = 0.7
x=2
--------------------------------------------------------------------------------------------------------------------This type of function is known as a STEP function. The reason for its name is obvious upon
viewing the graph of the function.
--------------------------------------------------------------------------------------------------------------------Graph the piecewise function on the coordinate plane to the right:
3, 0  x  1

g ( x)  6, 1  x  2
9, 2  x  3

---------------------------------------------------------------------------------------------------------------------
Try to write an equation in piecewise form for the step function h(x) graphed below:
Step functions are frequently used in real-life applications. One type of step function is called
the ceiling function. In a ceiling function, all non-integers are rounded up to the nearest
integer. An example of a ceiling function is when a phone service company charges by the
number of minutes used and always rounds up to the nearest integer of minutes. For example, if
your telephone call lasts 3.4 minutes, then you will be charged for a 4 minute phone call. The
ceiling function is denoted like so: f ( x)   x  .
Another type of step function is called the floor function. It is the opposite of the ceiling
function. In a floor function, all non-integers are rounded down to the nearest integer. An
example of a floor function would be one's age. If a person's age is technically 15 years and 4
months, then he/she will probably round down and say, "I am 15." A floor function is denoted
like so: g ( x)   x  . The floor function is also known as the greatest integer function. Its
representation is h( x)   x  .
Graph f ( x)   x  .
Graph g ( x)   x  .
--------------------------------------------------------------------------------------------------------------------A computer repair person charges $80 per hour for labor. However, she charges her labor in
increments of 15 minutes. For example, if she works for 39 minutes, she rounds up to 45
minutes and charges $60.
A)Write a function in piecewise form to represent the amount the repair person charges. It
should include all values between and including 0 and 90 minutes.
B)Is this function a type of ceiling function or floor function?
For Questions 1-4, graph the step functions on the coordinate planes.
1.
0, 0  x  3

g ( x)  1, 3  x  6
2, 6  x  9

3.
7, 0  x  1.5
5, 1.5  x  3

l ( x)  
3, 3  x  4.5
1, 4.5  x  6
2.
1,
h( x )  
1,
4.
2,

h( x)  1,
4,

x0
x0
1  x  1
2 x4
5 x7
For Questions 5-7, is the description a ceiling function or a floor function?
5.
When filling out your tax return, you do not have to include any cents on any of your
income nor expenses.
6.
In football, the statistician will give a running back credit for a 3-yard gain when he runs
for any gain greater than 2 yards but less than or equal to 3 yards.
7.
At the end of the semester, a nice teacher takes every student's average and raises it to the
first whole number score above their average. For example, an '83.2' would be an '84'.
For Questions 8-11, write the equation for the step function graphed on the coordinate plane.
8.
9.
6
6
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
10.
11.
6
6
4
4
2
2
-5
5
-5
5
-2
-2
-4
-4
--------------------------------------------------------------------------------------------------------------------12.
A lawyer charges a fee of $100 per hour of work. However, all charges are rounded to
the next hour up. So, if the lawyer works for 10 hours and 15 minutes on a case, then
the client is charged for 11 hours.
A)Write the equation for a function that shows the cost of work for any number of hours
greater than zero and no more than 5.
B)Is the function a ceiling function or a floor function?
13.
At the ticket counter at the local arcade, all prizes increase at increments of fifty tickets.
Thus, when Jerry takes all of his tickets to the counter, the number of tickets he possesses
over a multiple of 50 is irrelevant (does not matter).
A)Write the equation for a function that shows the number of important tickets as a
function of his actual number of tickets. The function should account for all ticket
numbers up to, but not including, 200 tickets.
B)Is this function a ceiling function or a floor function?