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Piecewise Functions and Step Functions

Piecewise Functions and Step Functions. 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.

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Piecewise Functions and Step Functions

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  1. Piecewise Functions and Step Functions

  2. 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.

  3. What do they look like? x2 + 1 , x  0 x – 1 , x  0 f(x) = You can EVALUATE piecewise functions. You can GRAPH piecewise functions.

  4. 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.

  5. II. Evaluating Piecewise Functions: Evaluating piecewise functions is just like evaluating functions that you are already familiar with. x2 + 1 , x  0 x – 1 , x  0 f(x) = 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

  6. Let’s calculate f(-2). x2 + 1 , x  0 x – 1 , x  0 f(x) = 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

  7. Your turn: 2x + 1, x  0 2x + 2, x  0 f(x) = Evaluate the following: f(-2) = -3 ? f(5) = 12 ? f(1) = 4 ? f(0) = ? 2

  8. One more: 3x - 2, x  -2 -x , -2  x  1 x2 – 7x, x  1 f(x) = Evaluate the following: f(-2) = 2 ? ? f(3) = -12 ? f(-4) = -14 ? f(1) = -6

  9. III. Graphing Piecewise Functions: x2 + 1 , x  0 x – 1 , x  0 f(x) = Determine the shapes of the graphs. Parabola and Line Determine the boundaries of each graph.   Graph the line where x is greater than or equal to zero. Graph the parabola where x is less than zero.          Notice the closed vs open circles.   Domain: Range:

  10. Graphing Piecewise Functions: 3x + 2, x  -2 -x , -2  x  1 x2 – 2, x  1 f(x) = Determine the shapes of the graphs. Line, Line, Parabola Determine the boundaries of each graph.                   

  11. 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.

  12. Notes: Step Functions

  13. 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.

  14. Try another:

  15. II. Special Step Functions • Two particular kinds of step functions are called ceiling functions ( f (x)= ]x[ and floor functions (f (x)=[x]). • 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.

  16. Least Integer Function:

  17. Least Integer Function:

  18. Least Integer Function:

  19. Least Integer Function:

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

  21. 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].

  22. Greatest Integer Function:

  23. Greatest Integer Function:

  24. Greatest Integer Function:

  25. Greatest Integer Function:

  26. 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.

  27. 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.

  28. 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.

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

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

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