Download Sec 1.3

Unit #1 - Transformation of Functions; Exponential and Logarithms
Section 1.3
Some material from “Calculus, Single and MultiVariable” 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.
TEST PREPARATION PROBLEMS
13. If f (x) = x2 + 1,
(a) f (t + 1) = (t + 1)2 + 1 = (t2 + 2t + 1) + 1 = t2 + 2t + 2
(b) f (t2 + 1) = (t2 + 1)2 + 1 = (t4 + 2t2 + 1) + 1 = t4 + 2t2 + 2
(c) f (2) = (2)2 + 1 = 5
(d) 2f (t) = 2(t2 + 1) = 2t2 + 2
(e) [f (t)]2 + 1 = [t2 + 1]2 + 1 = (t4 + 2t2 + 1) + 1 = t4 + 2t2 + 2
14. If f (n) = 3n2 − 2 and g(n) = n + 1,
(a) f (n) + g(n) = (3n2 − 2) + (n + 1) = 3n2 + n − 1.
(b) f (n)g(n) = (3n2 − 2)(n + 1) = 3n3 − 2n + 3n2 − 2 = 3n3 + 3n2 − 2n − 2
(c) Domain of f (n)/g(n): both f (n) and g(n) separately have domains of all real numbers. Therefore the only bad n values introduced by combining them are n values
for which the denominator, g(n), equals zero. g(n) = 0 when n = −1. Therefore the
domain of f (n)/g(n) is all real numbers except n = −1.
(d) f (g(n)) = f (n + 1) = 3(n + 1)2 − 2 = (3n2 + 6n + 3) − 2 = 3n2 + 6n + 1
(e) g(f (n)) = g(3n2 − 2) = (3n2 − 2) + 1 = 3n2 − 1
19. (a) In f (25), 25 represents the input, p, which is the price. f (25) represents the output,
or the number of items sold, given a price of $25.
(b) In f −1 (30), the input 30 is q, the number of items. Therefore, f −1 (30) represents the
price p at which 30 items would be sold.
20. (a) In f (10, 000), 10,000 represents the input A, the number of square feet. f (10, 000)
represents the cost of constructing a building with 10,000 square feet.
(b) In f −1 (20, 000), the input 20,000 is now a dollar amount, C. f −1 (20, 000) represents
the area of a building which would cost $20,000 to build.
21. f maps lengths x to temperatures o F . Therefore f −1 (75) maps a temperature (75o F ) to
the length of mercury (x, in inches).
22. (a) f (100) represents the gross income required to obtain a mortgage loan of $100,000.
(b) f −1 (75) represents the size of mortgage you could obtain if you had a gross income
of $75,000 per year.
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36. Function looks like a quadratic. It is flipped vertically, then shifted left by 1, and up by
3 units.
A possible formula would be y = 3 − (x + 1)2 .
37. This looks like a shift of x3 , with the point at the origin shifted to (2, -1). A possible
formula could be y = −1 + (x − 2)3 .
38. (a) The graph passes through (or close to) (-1, 2). This means that f −1 (2) ≈ −1.
(b)
39. The graph of Q = S(1 − e−kt ) = S − Se−kt , compared to Se−kt has had two changes:
minus sign in front of the main function, then +S. Thus, relative to the simpler function,
Q has been flipped/reflected vertically across y = 0, then shifted upwards by S.
59. A balloon has volume of V = 43 πr 3 . If r = 3t + 1, then at t = 3 the radius is 10. Using
3
this radius gives a volume of 43 π103 = 4000
3 π ≈ 4, 189 cm .
60. Height is y, and # of branches B = y − 1, and # of leaves per branch is n = 2B 2 − B.
The number of the leaves in the tree as a whole is (# of branches) × (# of leaves per
branch) = Bn = B(2B 2 − B).
Expressing this all in terms of the height y, Bn = (y − 1)(2(y − 1)2 − (y − 1)) = 2(y −
1)3 − (y − 1)2 . Simplification/expansion of this answer is not necessary.
61. (a) If C = 100 + 2q, then
C − 100 = 2q
C
C − 100
=
− 50 = q
2
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The inverse function is q = C2 − 50 = f −1 (C). Note: you cannot interchange q and
C because they represent real values in the problem (# of articles and cost).
(b) The formula tells you the number of articles q that could be produced for a total cost
C.
62. (a) Let p = an object’s mass in pounds, and k be its mass in kilograms. Then k = f (p) =
p/2.2.
(b) If f takes in mass in pounds, and outputs mass in kilograms, f −1 will take in a mass
in kg, and output a mass in pounds: p = f −1 (k) = 2.2k.
63. Removed from assignment; too vague.
64. (a) f (15) ≈ 45 or 50. This means that 15 million year-old rock can be found roughly
45 m beneath the ocean floor at this location.
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(b) f is invertible, because it passes the horizontal line test: each depth is associated
with just one time point.
(c) f −1 (120) ≈ 35. This means that, at a depth of 120 m below the ocean floor, you
will find rocks roughly 35 million years old.
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