piecewise and step functions

It’s time to put the FUN in FUNCTIONS!!!

Piecewise and Step 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 Functiony = [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, continuedNotice 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] continuedNotice 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 continuedNotice 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] + kThere 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 Step Functions on the CalculatorGo 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 Step Functions on the Calculator continuedNotice 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 WorldCell 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.

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 FunctionsIf , 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 = 9To find f(-2), we would use the first piece of

the function since -2 is less than 0.f(-2) = -(-2) = 2

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

The domain would be x < -2 and x 0. The range would be y -5.

The domain would be all real numbers. The range would be y > 4/5.

Piecewise Functions in the Real WorldThere 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.