Transcript Piecewise and Step Functions

Piecewise and Step Functions
It’s time to put the FUN in FUNCTIONS!!!
Lines…
 So far, you have done a lot of work with LINES.
 However, lines aren’t helpful when we are trying to model
real-world situations.
 In this lesson, we are going to begin to add more to your
“toolkit” of functions, so that you can begin to tackle more
relevant problems to life.
 ARE YOU READY??? 
The Greatest Integer Function
 y = [x] means “the greatest integer not greater than x”
 For example, [2.4], means “the greatest integer not greater
than 2.4.” So, [2.4] = 2.
 Here are some other examples:
 [3] = 3
 [-2.2] = -3
 [5.8] = 5
Greatest Integer Function, continued
 Notice that your answer will always be an integer.
 Because we can input any number in for x, but
only get integers as our output y, we need to look
at this graph because it probably looks VERY
different from all of the others we have studied so
far.
Let’s graph y = [x] using a table of
values:
x
y
-.5
-1
-.1
-1
0
0
0.4
0
0.8
0
1
1
1.2
1
1.8
1
2
2
2.3
2
3
3
3.2
3
3.7
3
Because this graph looks like a staircase,
we also call it a “step” function.
 Notice the open circle at the
end of each step. These occur
when we “jump” to the next
integer.
 The domain of y = [x] would
be all real numbers, because
we can substitute any number
in for x and get a result.
 The range of y = [x] would
be the set of integers, because
even though we can substitute
decimals or fractions, our
answer will always be an
integer.
Another Example: y = 2[x – 1]
x
y
0
-2
0.4
-2
0.8
-2
1
0
1.2
0
1.8
0
2
2
2.3
2
3
4
3.2
4
y = 2[x – 1] continued
 Notice the steps are further
apart this time: there are 2
spaces between each one.
 Also, the step at x = 0 has
shifted to the right 1 unit.
 The domain of y = 2[x – 1]
is all real numbers.
 The range of y = 2[x – 1] is
all even integers, because
we will only get even
integers as outputs.
Another Example: y = 0.5[x] + 2
x
y
0
2
0.4
2
0.8
2
1
2.5
1.2
2.5
1.8
2.5
2
3
2.3
3
3
3.5
3.2
3.5
y = 0.5[x] + 2 continued
 Notice the steps are closer
together this time: there is a
half of a space between each
one.
 Also, the step at x = 0 has
shifted up 2 units.
 The domain of y = 0.5[x] + 2
is all real numbers.
 The range of y = 0.5[x] + 2
is the set of numbers {…, 1.5, -1, -0.5, 0, 0.5, …},
because these are the
numbers we will get as
outputs.
Transformations of y = [x]
 General form: y = a[x – h] + k
 There will be a units between steps.
 The graph will shift h units right for [x – h]
and h units left for [x + h].
 The graph will shift up k units for +k and
down k units for –k.
Graphing The Greatest Integer Function
on the Calculator
 Go to y = as usual.
 Input your function. You
can find the greatest
integer function by going
to MATH, moving over to
NUM, and choosing #5
int(.
 For example, y = [x] + 2
would be put in as
y = int(x) + 2.
 Then, graph.
Graphing The Greatest Integer Function
on the Calculator continued
 Notice that it looks like the
steps are connected, which
we know is not the case.
This is because of your
calculator’s mode.
 Press MODE, and change
CONNECTED to DOT.
 Then, graph again. There
are our steps!
Step Functions in the Real World
 Cell phone plans: You pay one price for a specific number
of minutes. If you want more minutes, you pay more money.
Think of each step as a plan.
 The Post Office: You pay postage to mail things based on
their weight. One stamp allows you to mail something that
weighs up to17 ounces. Past the 17 ounces, you must add
another stamp, which means two stamps allows you to mail
up to 34 ounces. Think of each step as representing the
weight you can mail per stamp.
Step vs. Piecewise Function
 A step function is made up of pieces of a function that are in a
stair-step pattern.
 What do we call a function that is made up of pieces? We call
them piecewise functions.
 A step function is one type of piecewise function.
 The greatest integer function is one example of a step, but
not the only step function.
PIECEWISE
STEP
NOT STEP
GREATEST
NOT GREATEST
INTEGER
INTEGER
Piecewise Examples
 Click on each link to find out more about piecewise and
step functions
 Piecewise
http://www.mathsisfun.com/sets/functions-piecewise.html
 Step
http://www.icoachmath.com/math_dictionary/Step_Functio
n.html
Now let’s talk about piecewise
functions.
 Piecewise functions are defined on various intervals. In other
words, we will have different pieces of functions depending
on our x-values.
 For example:
 This means that we will graph f(x) = -x for all x-values less
than 0, and we will graph f(x) = 2x + 3 for all x-values
greater than or equal to 0. Let’s take a look at this graph.
More About Piecewise Functions
 If
, we could find functional
values for different values of x.
 For example, to find f(3), we would use the second piece of
the function since 3 is greater than or equal to 0.
f(3) = 2(3) + 3 = 9
 To find f(-2), we would use the first piece of the function
since -2 is less than 0.
f(-2) = -(-2) = 2
Graphing Piecewise Functions
1. Make a table for each “piece” of the function. The beginning of each
piece is the y-values. The second part if the values of x (domain).
2. Select x values in the given domain, starting with the endpoint.
3. If the domain (x) is > or <, use that endpoint but make it open. If the x is
≥ or ≤, make the endpoint a closed circle.
4. Graph all points and connect within the domain. The pieces may or may
not connect to each other.
 Let’s graph f(x) = -x,
where x must be less than
0.
 Let’s graph f(x) = 2x+3,
where x must be greater
than or equal to 0.\
 All together, here is the graph of our piecewise function:
 The domain would be all real numbers, and the range would be all
real numbers greater than 0.
Other Examples- Piecewise
x
-2
Open
-3
y=4
4
y = x-5
-5
4
x
0
closed
1
-4
4
-5
-6
x
y = -3 x+1
x
y=2
x
y =1/5x
1
2
4
open
5
4/5
4
0
open
1
2
-4
0
closed
-1
2
-3
-2
7
2
2
7
7/5
4
3
-2
-3
10
3
2
10
2
4
4
-1
-4
13
4
closed
2
15
3
The domain would be x < -2 and x
The range would be y -5.
0.
1
The domain would be all real numbers.
The range would be y > 4/5.
Piecewise Functions in the Real World
 There are so many ways that piecewise functions are used
every day.
 A t-shirt company sells t-shirts for $12 apiece if you buy 10
or less. If you are willing to buy between 11 and 50, they
will cut you a deal and sell them to you for $10 apiece. If
you are willing to buy more than 50, they will sell them to
you for $8 apiece.
Selling prices (y-values)
Number of t-shirts sold (x-values)
Piecewise Functions in the Real World
 A parking garage costs $2 per hour, with a maximum charge
of $12 for the day. This is an example of a step function.
0  x 1
y
20
1 x  2
2 x3
3 x  4
4 x5
5  x  24
18
16
14
Cost of Parking
2,
4,

6,
f ( x)  
8,
10,

12,
12
10
8
6
4
2
x
0
Parking cost $ (y-values) Parking time hrs (x-values)







Time in Parking Lot



