MHF4U Piecewise Functions

Some functions are represented by two or more “pieces.” These functions are called piecewise functions. Each piece of a piecewise function is defined for a specific interval in the domain of the function. Example 1 A city parking lot uses the following rules to calculate parking fees: A rate of $2 up to and including the first hour An additional $2 for every hour up to and including 5 hours A flat rate of $10 for any amount of time after 5 hours (up to 12 hours) Representing the problem using a graphical model 26955757302557150133985Time (h) Parking Fee ($) 0 0.25 0.5 0.75 1.00 1.25 1.50 2.00 3.00 4.00 5.00 6.00 00Time (h) Parking Fee ($) 0 0.25 0.5 0.75 1.00 1.25 1.50 2.00 3.00 4.00 5.00 6.00 Representing the problem using an algebraic model How do you determine from a graph or from the algebraic representation of a piecewise function if there are any discontinuities? 3813810381000Example 2 Graph the following piecewise function. fx=x2, x<22x+3, x?2 Algebraically show that this function is discontinuous. Thinking 200406067627500Example 3 Graphically create a continuous piecewise function, using 3 different base functions (including a sinusoidal function as one of them). Then write the equation of the function. Piecewise Function Worksheet 1. Graph the following piecewise functions. Determine whether each function is continuous or discontinuous, and state he domain and range of the function. a) b) c) d) 2. Write the algebraic representation of each piecewise function, using function notation. 360997571120 00 31432599695 00 a) b) 3. Graham’s long-distance telephone plan includes the first 500 min per month in the $15.00 monthly charge. For each minute beyond 500 min, Graham is charged $0.02. Write a function that describes Graham’s total long-distance charge in terms of the number of long distance minutes he uses in a month. 4. Many income tax systems are calculated using a tiered method. Under a certain tax law, the first $100 000 of earnings are subject to a 35% tax; earnings greater than $100 000 and up to $500 000 are subject to a 45% tax. Any earnings above $500 000 are taxed at 55%. Write a piecewise function that models this situation. 5. Find the value of k that makes the following function continuous. Graph the function. 6. The fish population, in thousands, in a lake at any time, x, in years is modeled by the following function: This function describes a sudden change in the population at time x=6, due to a chemical spill. a) Sketch the graph of the piecewise function. b) Describe the continuity of the function. c) How many fish were killed by the chemical spill? d) At what time did the population recover to the level it was before the chemical spill? e) Describe other events relating to fish populations in a lake that might result in piecewise functions. Answers 1609725125730Discontinuous at x=-1 00Discontinuous at x=-1 4591050116205Continuous 00Continuous 341947549530 00 4857751440180 00 49530059055 00 1.a) b) 1743075175895Continuous 00Continuous 4648200137795Continuous 00Continuous 340042590170 00 c) d) 2.a b) 3. 4. 5. 210502552705b. The function is discontinuous at x=6 c. 32 000 fish d. 8 years after the spill e. Answers may vary. For example, three possible events are: environmental changes, introduction of a new predator, and increased fishing. 00b. The function is discontinuous at x=6 c. 32 000 fish d. 8 years after the spill e. Answers may vary. For example, three possible events are: environmental changes, introduction of a new predator, and increased fishing. 35242552705 00 6.