a Sketch the function and determine whether it is invertible.. b If the function is invertible, sketch the inverse function on the same set of axes as the function and find a formula for
Trang 1f (x)
f–1(x)
2 2
Figure 12.6
This makes sense: The function f cubes its input, multiplies the result by 4, and then adds
2 To undo this, we must subtract 2, divide the result by 4, and then take the cube root See the diagram below for clarification
x
cube
multiply by 4
x3
4x3
cube root
divide by 4 f–1
f
subtract 2
4x3 +2 add 2
EXAMPLE 12.6 Let g(x) = x2 Find g−1(x)if g is invertible
At first glance, it may seem that if g is the squaring function, its inverse must be the square root function But we must be careful g is not invertible because it is not 1-to-1 The problem is that given any positive output, say 4, it is impossible to determine uniquely the corresponding input The input corresponding to 4 could be 2 or −2
Trang 2432 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone?
x
y
g (x)
g is NOT invertible
Figure 12.7
Had we not noticed this and proceeded to look for the inverse analytically, we would soon realize that there was a problem
Set y = x2and then interchange the roles of x and y to obtain an inverse relationship
x = y2 Solve for y
y = ±√x yis not a function of x
What we can do is restrict the domain of g to make g 1-to-1 If we restrict the domain to [0, ∞), then g−1(x) =√x If we restrict the domain to (−∞, 0], then g−1(x) = −√x
x
x
g (x) = x2 g (x) = x2
g–1(x) = √x
g–1(x) = – √x
(i) g(x) on [0, ∞)
g–1(x) = √x
(ii) g(x) on (–∞, 0]
g–1(x) = – √x
Figure 12.8
Observation
If a function is not 1-to-1 on its natural domain it is possible to restrict the domain in order
to make the function invertible Note that the domain of f is the range of f−1and the range
of f is the domain of f−1
P R O B L E M S F O R S E C T I O N 1 2 2
1 For each of the functions f , g, and h below, do the following
(a) Sketch the function and determine whether it is invertible
(b) If the function is invertible, sketch the inverse function on the same set of axes as the function and find a formula for the inverse function
Trang 3i f (x) =√x − 1 ii g(x) =x1 iii h(x) =x3 + 1
2 For each of the functions below, find f−1(x)
(a) f (x) = 2 −x + 1x (b) f (x) =x105 + 7
3 Suppose f is an invertible function
(a) If f is increasing, is f−1increasing, decreasing, or is there not enough information
to determine?
(b) If f is decreasing, is f−1increasing, decreasing, or is there not enough information
to determine?
(c) Suppose f is increasing and concave up Is f−1 concave up or concave down?
(Hint: Let y = f (x) What happens to the ratio *y*xas x increases? How does this translate into information about the inverse function? Check your conclusion with
a concrete example.) We will be able to work this out analytically by Chapter 16
4 Let
f (x) =2x − 1 3x + 4. Find f−1(x)
5 The function f is increasing and concave up on (−∞, ∞) f(x)is never zero Denote
by g(x) the inverse of f
(a) What is the sign of g?
(b) What is the sign of g?
(c) If f (3) = 5 and f(3) = 10, what is g(5)?
The functions in Problems 6 through 10 are 1-to-1 Find f−1(x) and specify the domain
of f−1.
6 f (x) =x+3x
7 f (x) =3−x2
8 f (x) =√x + 3
9 f (x) = 2√x − 6
10 f (x) = x3+ 1
For Problems 11 through 16, use the first derivative to determine whether the function given is 1-to-1 If it is, find its inverse function.
11 f (x) = x3+ 2x − 3
12 f (x) = x3− 2x + 3
Trang 4434 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone?
13 f (x) = |x − 3|
14 f (x) = x2+ 2x − 1
15 f (x) = 3 · 2x
16 f (x) = 5 · 3−x
12.3 INTERPRETING THE MEANING
OF INVERSE FUNCTIONS
In order to make sense of the information given by an inverse function in some real-world context it is helpful to clarify in words the input and output of both the function and its inverse We illustrate this in the example below
EXAMPLE 12.7 Let P (t) be the amount of money in a bank account at time t, where t is measured in years
and t = 0 represents January 1, 1998 Suppose that P0dollars are originally deposited in the account
Interpret the following expressions in words
(a) P (2) (b) P−1(5000) (c) P−1(2P0) (d) P−1(2P0+ 10) (e) P−1(2P0) + 10
SOLUTION Begin by firmly establishing the input and output of P and P−1
P
P–1
P(time) = dollars, so P−1(dollars) = time
(a) P (2) is the number of dollars in the account at t = 2 (January 1, 2000).
(b) P−1(5000) is the time when the balance will be $5000
(c) P−1(2P0)is the time when the balance will be twice P0, i.e., when the balance will have doubled from its original amount
(d) P−1(2P0+ 10) is the time when the balance will be $10 more than twice the original
balance
(e) P−1(2P0) + 10 is the time ten years after the balance is twice P0, i.e., ten years after the balance has doubled
P R O B L E M S F O R S E C T I O N 1 2 3
1 Let C(q) be the cost (in dollars) of producing q items Translate the following equations into words
Trang 5(b) C−1(1000) = 500
(c) C(200) = 1.5
2 Apricots are sold by weight In other words, the price is proportional to the weight Let C(w)be the cost of w pounds of apricots Suppose that A pounds of apricots cost $3 (a) Describe in words the practical meaning of each of the following and then evaluate the expression (When evaluating, use the fact that price is proportional to weight Your answers should be either a number or an expression in terms of A.)
i C(3A)
ii C−1(6)
iii C−1(1)
(b) In this particular situation, which of the following statements are true?
i C(3A) = 3C(A)
ii C−1(2x) = 2C−1(x)
iii C−1(x2) =C−12(x)
iv C−1(x + x) = C−1(2x)
(c) Only one of the statements above is true for any invertible function C Which
statement is this?
3 Let C(q) be the cost of producing q items Suppose that right now A items have been produced at a cost of $B Interpret the following expressions in words “A” and “B” should not appear in your answers; use words instead
(a) C(400)
(b) C−1(3000)
(c) C−1(B + 100)
(d) C(A + 10)
(e) C−1(2B)
4 A typist’s daily wages are determined by the number of words per minute he averages
on his shift Let D(w) be his daily earnings (in dollars) as a function of w, the average number of words per minute he types Suppose that yesterday he was paid $B for averaging C words per minute
Interpret each of the following equations or expressions in words Your answer should be expressed in terms of pay and words per minute
(a) D−1(70) = 50
(b) D(C + 5) = 1.1B
(c) D−1(B + 10)
5 Let R(d) be a function that models a company’s annual revenue (the amount of money they receive from customers) in dollars as a function of the number of dollars they spend that year on advertising Suppose that last year they spent $B on advertising and took in a total revenue of $C
Interpret each of the following equations or expressions Your answers should not contain $C or $B, but words instead
Trang 6436 CHAPTER 12 Inverse Functions: Can What Is Done Be Undone?
(a) R(B/2) = C − 80,000 (b) R(30,000) = 2.8 (c) R−1(2C)
6 Let f (t) = 5(1.1)6t+2+ 1
(a) The point (13, 6) lies on the graph of f What is f−1(6)?
(b) Find a formula for f−1(t )
(c) Use your formula to find f−1(6) Does your answer agree with your answer to part (a)?
(d) If f (t) models the number of pounds of garbage in a garbage dump t days after the dump has officially opened, interpret f−1(30) in words
7 A ball is thrown straight up into the air t seconds after it is released, its height is given
by H (t) = −16t2+ 96t feet
(a) Sketch a graph of H (t)
(b) What is the domain of H (t)? The range?
(c) What is the ball’s maximum height? When does it attain this height?
(d) Sketch the inverse relation for H (t) Is it a function? Explain
(e) How can you restrict the domain of H (t) so that it will have an inverse?
(f ) Having restricted the domain so that H (t) is invertible, evaluate H−1(80) What
is its practical meaning?
Trang 7Thinking About the Derivatives of Inverse Functions
1.Let f (x) = x2 with the domain of f restricted to x ≥ 0 Then
f−1(x) =√x
(a) Compare the derivative of x2at (3, 9) with the derivative of
√
x at (9, 3) Accompany your answer by a sketch
(b) Compute the derivative of x2at (a, b)
(c) Compute the derivative of√x
at (b, a)
(d) Compare your answers to the previous two questions by either
expressing both in terms of a or expressing both in terms of
b How are they related?
2.The function exis invertible Denote its inverse function by g(x)
(a) On the same set of axes, graph exand its inverse function g(x)
What is the domain of g? The range of g?
(b) The derivative of ex at (0, 1) is 1 What do you think the
derivative of the inverse function is at (1, 0)?
(c) What can you say about the sign of gand of g?
Trang 9Logarithmic Functions
13.1 THE LOGARITHMIC FUNCTION DEFINED
Introductory Example
EXAMPLE 13.1 A large lake has been serving as a reservoir for its nearby towns Over the years, industries
on the shore of the lake have contributed to the pollution of the lake An awareness of the problem has caused community members to ban further pollution Due to a combination of runoff and natural processes, the amount of pollutants in the lake is expected to decrease at
a rate proportional to pollution levels The number of grams of pollutant in the lake is now 1200; t years from now the number of grams is expected to be given by 1200(10)−t/8 If the water is deemed safe to drink only when the pollutant level has dropped to 400 grams, for how many years will the towns need to find an alternative source of drinking water?
SOLUTION We must find t such that 400 = 1200(10)−t/8 This is equivalent to solving1200400 = (10)−t/8,
or13= (10)−t/8 We can approximate the solution using a graphing calculator One approach
is to look for the root of 13− (10)−t/8 Another is to look for the point of intersection of
y = 1200(10)−t/8and y = 400
But suppose we would like an exact answer; we want to solve the equation analytically for t We could simplify somewhat by converting13= (10)−t/8to13= (10t)−1/8and raising both sides of this equation to the (−8) to get13−8= 10t We know13−8=(3)−1−8= (3)8= 6561, so we must solve the equation
10t= 6561
tis the number we must raise 10 to in order to get 6561 Since 103= 1000 and 104= 10,000,
we can be sure that t is a number between 3 and 4 Again we could revert to our calculator
to get better and better estimates of the value of t However, if we can find the inverse of the
439
Trang 10440 CHAPTER 13 Logarithmic Functions
function 10t, then we can find the exact solution to the equation 10t
= 6561 If f (t) = 10t, then t = f−1(6561)
The Inverse of f (x) = 10x
We know the function f (x) = 10xis invertible because it is 1-to-1 The inverse function,
f−1, is obtained by interchanging the input and output of f ; the graph of f−1can be drawn
by reflecting the graph of 10xover the line y = x
x y
y = x
f –1(x)
f (x) = 10 x
1 1
Figure 13.1
Suppose we go about looking for a formula for f−1in the usual way, by interchanging the roles of x and y and solving for y We write x = 10y What is y? y is the number we must raise 10 to in order to get x We don’t have an algebraic formula for this, but this function
is quite useful, so we give it a name
D e f i n i t i o n
log10xis the number we must raise 10 to in order to obtain x log10x is often written log x We read log10xas “log base 10 of x.”
y = log10xis equivalent to 10y= x
The domain of f is (−∞, ∞) and the range is (0, ∞) Therefore the domain of log x
is (0, ∞) and its range is (−∞, ∞) Note then that log x is defined only for x > 0 By examining the graph in Figure 13.1, we see that the graph of log x is increasing and concave
down; it is increasing without bound, but it is increasing very slowly.
lim
x→0 +log10x = −∞ lim
x→∞log10x = +∞
Note that although we now have a nice compact way of expressing the inverse function
of 10x, it might seem that all we have really accomplished so far is the introduction of
a shorthand for writing “the number we must raise 10 to in order to obtain x.” But there
is a definite perk A calculator will give a numerical estimate of the logarithm up to 10