Piecewise Functions and Step Functions I What Are

Piecewise Functions and Step Functions

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

What do they look like? f(x) = x 2 + 1 , x 0 x– 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 person, while groups of 50 people or more are charged a reduced rate of 30. 00 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. Let’s calculate f(2). f(x) = x 2 + 1 , x 0 x – 1 , x 0 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

Let’s calculate f(-2). f(x) = x 2 + 1 , x 0 x – 1 , x 0 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) = 2 x + 1, x 0 2 x + 2, x 0 Evaluate the following: f(-2) = -3? f(5) = 12 ? 2? f(1) = 4? f(0) =

One more: 3 x - 2, x -2 -x , -2 x 1 x 2 – 7 x, x 1 f(x) = Evaluate the following: 2? f(3) = ? -12 ? f(-4) = -14 f(1) = -6? f(-2) =

III. Graphing Piecewise Functions: f(x) = x 2 + 1 , x 0 x – 1 , x 0 Determine the shapes of the graphs. Parabola and Line Determine the boundaries of each graph. Graph the line where Graph the parabola x is greater than or where x is less than equal to zero. Notice the closed vs open circles. Domain: Range:

Graphing Piecewise Functions: 3 x + 2, x -2 f(x) = -x , -2 x 1 x 2 – 2, x 1 Determine the shapes of the graphs. 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 person, while groups of 50 people or more are charged a reduced rate of 30. 00 person. • Find a mathematical model expressing the amount a group will be charged for admission as a function of its size.

Notes: Step Functions

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.

Try another:

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 nearest integer. 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 Function:

Least Integer Function:

Least Integer Function:

Least Integer Function: 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 Function In 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 Function:

Greatest Integer Function:

Greatest Integer Function:

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