Week #1 The Exponential and Logarithm Functions Section 1.3 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. SUGGESTED PROBLEMS 15. If f(x) = x 2 + 1, find and simplify: (a) f(t + 1) (b) f(t 2 + 1) (c) f(2) (d) 2f(t) (a) f(t + 1) = t 2 + 2t + 2 (b) f(t 2 + 1) = t 4 + 2t 2 + 2 (c) f(2) = 5 (d) 2f(t) = 2t 2 + 2 16. For f(n) = 3n 2 2 and g(n) = n + 1, find and simplify: (a) f(n) + g(n) (b) f(n)g(n) (c) The domain of f(n)/g(n) (d) f(g(n)) (e) g(f(n)) (a) f(n) + g(n) = 3n 2 + n 1 (b) f(n)g(n) = 3n 3 + 3n 2 2n 2 (c) Domain of f(n)/g(n) wherever g(n) 0, which mean for all n 1 (d) f(g(n)) = 3n 2 + 6n + 1 (e) g(f(n)) = 3n 2 1 21. Let p be the price of an item and q be the number of items sold at that price, where q = f(p). What do the following quantities mean in terms of prices and quantities sold? (a) f(25) (b) f 1 (30) (a) f(25) is the number of items sold when the price is p = 25. (b) f 1 (30) is the price at which q = 30 items would be sold. 1
For Exercises 25-26, decide if the function y = f(x) is invertible. 25. Fails the horizontal line test, so this function is not invertible. 26. Fails the horizontal line test, so this function is not invertible. 27. (a) Use Figure 1.32 to estimate f 1 (2). (b) Sketch a graph of f 1 on the same axes. Figure 1.32 (a) For f 1 (2), we are looking for the point where the graph reaches the value y = 2. This occurs near x = 1, so f 1 (2) 1. (b) Reflect the graph across the 45 o, y = x line. QUIZ PREPARATION PROBLEMS Find possible formulas for the graphs in Exercises 28-29 using shifts of x 2 or x 3. 28. 2
y = (x + 1) 2 + 3 29. y = (x 2) 3 1 Are the functions in Exercises 30-37 even, odd, or neither? 30. f(x) = x 6 + x 3 + 1 Neither (combination of even power x 6 and odd power x 3 ). Check that f( x) f(x) and f(x) 31. f(x) = x 3 + x 2 + x Neither (combination of even power x 3 and odd power x 2 ). Check that f( x) f(x) and f(x) 32. f(x) = x 4 x 2 + 3 Even. If we replace x with x in all terms, the even powers will change them back into xs, or f( x) = ( x) 4 ( x) 2 + 3 = x 4 x 2 + 3 = f(x) 33. f(x) = x 3 + 1 Neither, as the constant acts as an even function. (Or, an odd function must pass through (0,0)). If we find f( x), we see f( x) = ( x) 3 + 1 = x 3 + 1 f(x) and f(x) 34. f(x) = 2x Odd. f( x) = 2x = f(x) 35. f(x) = e x2 1 Even. f( x) = e ( x)2 1 = e x2 1 = f(x) 36. f(x) = x(x 2 1) Odd. f( x) = ( x)(( x) 2 1) = x(x 2 1) = f(x) 37. f(x) = e x x Neither. f( x) = e x + x f(x) and f(x) 38. (a) Write an equation for a graph obtained by vertically stretching the graph of y = x 2 by a factor of 2, followed by a vertical upward shift of 1 unit. Sketch it. 3
(b) What is the equation if the order of the transformations (stretching and shifting) in part (a) is interchanged? (c) Are the two graphs the same? Explain the effect of reversing the order of transformations. (a) y = 2x 2 + 1 y 0 2 4 6 8 0.0 0.4 0.8 1.2 x (b) y = 2(x 2 + 1) y 0 2 4 6 8 2 1 0 1 2 x (c) The two graphs are clearly not the same. Besides the change in the resulting steepness of the curve, the y axis intercept in (a) is 1 (only has the final vertical shift), while in (b) the vertical shift is emphasized by the later vertical scaling. For Problems 39-42, decide if the function f is invertible. 39. f(t) is the number of customers in Macy s department store at t minutes past noon on December 18, 2000. 4
Not invertible. Given a certain number of customers, say f(t) = 1500, there could be many times, t, during the day at which that many people were in the store. So, we don t know which time instant is the right one. 40. f(n) is the number of students in your calculus class whose birthday is on the nth day of the year. Not invertible. If the number of students in the class is less than 363, then there have to be at least two days when no one has a birthday.this means that f(n) = 0 for two different values of n which guarantees that f is not invertible. If the class has 364 or more students, the argument above does not work. However,it is not hard to see that in order for f(n) to be different for every value of n, we need 365 different numbers to assign to the 365 different days of the year. Since f(1) + f(2) +... + f(365) = the number of students in the class, the smallest class size that allows every day to have a number of student birthdays that is different from the number on any other day, is 0 + 1 +... + 364 = (364)(365)/2 = 66,430. This is much too large to be the number of students in a class (even MATH 121!) 41. f(x) is the volume in liters of x kg of water at 4C. Invertible. Since at 4 o C, the mass of one liter of water is 1 kilogram, the mass of x liters is x kilograms. So, f(x) = x and therefore, f 1 (x) = x. 42. f(w) is the cost of mailing a letter weighing w grams. Not invertible, since it costs the same to mail a 50-gram letter as it does to mail a 51-gram letter. For Problems 43-48, use the graphs in Figure 1.33. Figure 1.33 43. Estimate f(g(1)) f(g(1)) = f(2) 0.4 45. Estimate f(f(1)) f(f(1)) f( 0.5) 1 52. The cost of producing q articles is given by the function C = f(q) = 100 + 2q. (a) Find a formula for the inverse function. 5
(b) Explain in practical terms what the inverse function tells you. (a) The function f tells us C in terms of q. To get its inverse, we want q in terms of C, which we find by solving for q: C = 100 + 2q C 100 = 2q q = C 100 = f 1 (C) 2 (b) The inverse function tells us the number of articles that can be produced for a given cost. 54. Complete the following table with values for the functions f, g, and h, given that: (a) f is an even function. (b) g is an odd function. (c) h is the composition h(x) = g(f(x)) x f(x) g(x) h(x) -3 0 0-2 2 2-1 2 2 0 0 0 1 2 3 x f(x) g(x) h(x) -3 0 0 0-2 2 2-2 -1 2 2-2 0 0 0 0 1 2-2 -2 2 2-2 -2 3 0 0 0 6