A Piecewise Defined Function

Transcription

1 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Eample A Piecewise Defined Function A function may employ different formulas on different parts of its domain. Such a function is said to be piecewise defined. For eample, the function graphed has the following formulas: y = 2 y y = 6 - y 2 6 for for 2 2 Label (major) Endpoints!!!

2 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally The Absolute Value Function The Absolute Value Function is defined by f ( ) for for 0 0

3 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Graph of y = (-3,3) y (3,3) (-2,2) (2,2) (-1,1) (1,1) (0,0) The Domain is all real numbers, The Range is all real nonnegative numbers.

4 Tet Eample 3 The Ironman Triathlon is a race that consists of three parts: a 2.4 mile swim followed by a 112 mile bike race and then a 26.2 mile marathon. A participant swims steadily at 2 mph, cycles steadily at 20 mph, and then runs steadily at 9 mph. Assuming that no time is lost during the transition from one stage to the net, find a formula for the distance d, covered in miles, as a function of the elapsed time t in hours, from the beginning of the race. Then Graph the function.

5 Tet Eample 3 d 2t 20t RUN 21.6 BIKE for for t t 53.2 SWIM for

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8 Show how the equations and the end points generated? d 2t 20t 21.6 for for t t 53.2 for

9 The charge for a tai ride in New York City is $2.50 for the first 1/4 of a mile, and $0.40 for each additional ¼ of a mile (rounded up to the nearest 1/4 mile). (a) Make a table showing the cost of a trip as a function of its length. The table should start at zero and increase to two miles in 1/4-mile intervals.

10 (b) What is the cost for a 1.25-mile trip? (c) How far can you go for $5.30? (d) Graph the cost function in part (a).

11 Graph the piecewise function f ( ) for for for 7 100

12 88-21 The rate at which people enter an office building is given in the graph following. A negative rate means that people are leaving the building.

13 RATE 0 A B C D E F G H I J TIME K

14 1) Write appropriate units for each of the variables.

15 2) Create reasonable coordinate values for the 12 blue points indicated on the graph.

16 For each of the following statements give the largest interval on which:

17 (a) The number of people in the building is increasing.

18 (b) The number of people in the building is constant.

19 (c) The number of people in the building is increasing fastest.

20 (d) The number of people in the building is decreasing.

21 HW: Using the values you created, generate formulas for each of the 10 straight lines in the graph.

22 Suppose w = j () is the average daily quantity of water (in gallons) required by an oak tree of height feet. (a) What does the epression j(25) represent? What about j -1 (25)?

23 (b) What does the following equation tell you about v: j (v) = 50. Rewrite this statement in terms of j -1 (c) Oak trees are on average z feet high and a tree of average height requires p gallons of water. Represent this fact in terms of j and then in terms of j -1 (d) Using the definitions of z and p from part (c), what do the following epressions represent (INTERPRET)?

24 j(2z) 2 j( z) j( z 10) j( z) 10 j 1 (2 p) j 1 ( z 10) j 1 ( z) 10

25 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Inverse Functions The roles of a function s input and output can sometimes be reversed. For eample, the population, P, of birds on an island is given, in thousands, by P = f(t), where t is the number of years since In this function, t is the input and P is the output. If the population is increasing, knowing the population enables us to calculate the year. Thus we can define a new function, t = g(p), which tells us the value of t given the value of P instead of the other way round. For this function, P is the input and t is the output. The functions f and g are called inverses of each other. A function which has an inverse is said to be invertible.

26 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Inverse Function Notation If we want to emphasize that g is the inverse of f, we call it f 1 (read f-inverse ). To epress the fact that the population of birds, P, is a function of time, t, we write P = f(t) To epress the fact that the time t is also determined by P, so that t is a function of P, we write t = f 1 (P) The symbol f 1 is used to represent the function that gives the output t for a given input P. Warning: The 1 which appears in the symbol f 1 for the inverse function is not an eponent.

27 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Finding a Formula for the Inverse Function Eample 6 The cricket function, which gives temperature, T, in terms of chirp rate, R, is T = f(r) = ¼ R Find a formula for the inverse function, R = f 1 (T ). Solution The inverse function gives the chirp rate in terms of the temperature, so we solve the following equation for R: T = ¼ R + 40, giving T 40 = ¼ R, R = 4(T 40). Thus, R = f 1 (T ) = 4(T 40).

28 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Domain and Range of an Inverse Function Functions that possess inverses have a oneto-one correspondence between elements in their domain and elements in their range. The input values of the inverse function f 1 are the output values of the function f. Thus, the domain of f 1 is the range of f. Similarly, the domain of f is the range of f -1

29 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally A Function and its Inverse Undo Each Other The functions f and f 1 are called inverses because they undo each other when composed: f (f -1 ()) = f (f -1 ()) = Eample 7 Given T = f(r) = ¼ R + 40 R = f 1 (T ) = 4(T 40), f (f -1 (T)) = ¼ (4(T 40)) + 40 = T = T and f -1 f(r)) = 4( ¼ R ) = R = R