piecewise functions and step functions. i.what are they? up to now, we’ve been looking at...

Piecewise Functions and Step Functions

I.I. What Are They?What Are They?

Up to now, we’ve been looking at functions represented by a single equation.

In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain.

These are called piecewise functions.

Piecewise Function Piecewise Function –a function defined by two or –a function defined by two or more functions over a specified domain.more functions over a specified domain.

What do they look like?

f(x) = x2 + 1 , x 0x – 1 , x 0

You can EVALUATE piecewise functions.

You can GRAPH piecewise functions.

When do we use them in real life?• All the time.• Here is one example:• 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.

• This situation can be represented by a piecewise function. We will come back to this example at the end of the lesson.

II. Evaluating Piecewise Functions:

Evaluating piecewise functions is just like evaluating functions that you are already familiar with.

f(x) = x2 + 1 , x 0x – 1 , x 0

Let’s calculate f(2).

You are being asked to find y when x = 2. Since 2 is 0, you will only substitute into the second part of the function.

f(2) = 2 – 1 = 1

f(x) = x2 + 1 , x 0x – 1 , x 0

Let’s calculate f(-2).

You are being asked to find y when x = -2. Since -2 is 0, you will only substitute into the first part of the function.

f(-2) = (-2)2 + 1 = 5

Your turn:

f(x) = 2x + 1, x 02x + 2, x 0

Evaluate the following:

f(-2) = -3?

f(0) = 2?

f(5) = 12?

f(1) = 4?

One more:

f(x) = 3x - 2, x -2-x , -2 x 1x2 – 7x, x 1

Evaluate the following:

f(-2) = 2?

f(-4) = -14

?

f(3) = -12

?

f(1) = -6?

III. Graphing Piecewise Functions:

f(x) = x2 + 1 , x 0x – 1 , x 0

Determine the shapes of the graphs.

Parabola and LineDetermine the boundaries of each graph.

Graph the parabola where x is less than zero.

Graph the line where x is greater than or equal to zero.

Notice the closed vs open circles.

Domain: Range:

3x + 2, x -2-x , -2 x 1x2 – 2, x 1

f(x) =

Graphing Piecewise Functions:

Determine the shapes of the graphs.Line, Line, Parabola

Determine the boundaries of each graph.

IV. Applications

• 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.

• Find a mathematical model expressing the amount a group will be charged for admission as a function of its size.

Notes: Step Functions

43 ,432 ,321 ,210 ,1

)(

xifxifxifxif

xf

I. What is it?

• A step function looks like a steps on a staircase. They can be represented by a piecewise function, or the greatest integer function. Try graphing the following piecewise function.

43,432,321,210,1

)(

xifxifxifxif

xf

Try another:

01,412,323,234,1

)(

xifxifxifxif

xf

II. Special Step Functions

Two particular kinds of step functions are called ceiling functions ( f (x)= ]x[ and floor functions (f (x)=[x]).

A.Ceiling Functions:In a ceiling function, all nonintegers are rounded up to the nearestinteger. This is also called the ‘least integer function’.

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.

Least Integer Function:

least integer that is xy

x y

0 00.5 1

0.75 11 1

x y

0 00.5 1

0.75 11 1

1.5 22 2



The least integer function is also called the ceiling function.

The notation for the ceiling function is:



The TI-89 command for the ceiling

function is ceiling (x).

Don’t worry, there are not wall functions, front door functions, fireplace functions!

B. Floor Function/Greatest Integer FunctionIn a floor function, all nonintegers are rounded down to the nearest integer.

The way we usually count our age is an example of a floor function since we round our age down to the nearest year and do not add a year to our age until we have passed our birthday.

The floor function is the same thing as the greatest integer function which can be written as f (x)=[x].

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

Greatest Integer Function:

greatest integer that is xy

x y

0 00.5 0

0.75 01 1

1.5 12 2

III. Applications of Step Functions

PSYCHOLOGY One psychologist charges for counseling sessions at the rate of $85 per hour or any fraction thereof. Draw a graph that represents this situation.

Understand The total charge must be a multiple of $85, so the graph will be the graph of a step function.

Plan If the session is greater than 0 hours, but less than or equal to 1 hour, the cost is $85. If the time is greater than 1 hour, but less than or equal to 2 hours, then the cost is $170, and so on.

Solve Use the pattern of times and costs to make a table, where x is the number of hours of the session and C(x) is the total cost. Then draw the graph.

Answer:

Check Since the psychologist rounds any fraction of an hour up to the next whole number, each segment on the graph has a circle at the left endpoint and a dot at the right endpoint.

Try this!SALES The Daily Grind charges $1.25 per pound of meat or any fraction thereof. Draw a graph that represents this situation.

A. B.

C. D.

Homework

• You are to complete #60 and #61 tonight. • #60: Graphing Piecewise Functions• Skip #2• #61 Step Functions WS• Skip # 5

piecewise functions and step functions. i.what are they? up to now, we’ve been looking at...

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