If a function acts on its input by adding 3, the action can be undone by subtracting 3.. If a function acts by doubling its input, the action can be undone by halving.. If f is a functio
Trang 1IV Inverse Functions: A Case Study of Exponential and Logarithmic Functions
12
Inverse Functions: Can What Is Done Be Undone?
What’s done cannot be undone
—MacBeth, Act V, scene 1
Now mark me, how I will undo myself
—Richard II, Act IV, scene 1
12.1 WHAT DOES IT MEAN FOR F AND G TO BE
INVERSE FUNCTIONS?
Some actions can be undone; other actions, once taken, can never be undone If we think
of a function as an action on an input variable to produce an output, we can make a similar observation Let’s begin by looking at functions whose actions can be undone If a function acts on its input by adding 3, the action can be undone by subtracting 3 If a function acts
by doubling its input, the action can be undone by halving If f is a function whose action
can be undone, we refer to the function that undoes the action of f as its inverse function
and denote it by f−1 We read f−1as “f inverse.”
CAUTIONThis notation, while quite standard, can be very misleading “f inverse” is not
1
f (x) The reciprocal of f (x) is writtenf (x)1 or [f (x)]−1
421
Trang 2EXAMPLE 12.1 Below are some simple functions together with their inverse functions.
Function Action done Action undone Inverse function
i f (x) = x + 3 Add 3 to input Subtract 3 from input f−1(x) = x − 3
ii g(x) = 2x Double input Halve input g−1(x) = x/2 iii h(x) = x3 Cube the input Take the cube root h−1(x) = x1/3
Notice again that the inverse of f is not the reciprocal1of f
What exactly do we mean when we say “the inverse function of f undoes the action of
f”? If f assigns to the input “a” the output “b,” then its inverse, f−1, assigns to the input
“b” the output “a.” That is, if f (a) = b, then f−1(b) = a Equivalently, for every point (a, b) on the graph of f , the point (b, a) lies on the graph of f−1
input of f
output of f–1
f–1
f
y1
y2
x2
x1
x3
y3
a→ bf f→ a−1
If f and f−1are inverse functions, f−1undoes the action of f and vice versa, so
a→ bf f→ a and b−1 f→ a−1 → bf
input of f
f(a) = b
input of f–1
output of f–1
f–1(b) = a
output of f
Figure 12.1
Look at Example 12.1(i), where f (x) = x + 3 and f−1(x) = x − 3
4→ 7f f→ 4−1 and 7f→ 4−1 → 7f
5f→ 2−1 → 5f and 2→ 5f f→ 2−1 More generally, imagine sending x down an assembly line The first machine on the line, let’s call it f , acts on x Its output, f (x), is passed along to the second machine, f−1, which undoes the work of the first Therefore, the final output is x
x→ f (x)f f→ x−1 and similarly xf→ f−1 −1(x)→ x.f
1
Do not confuse inverse functions with the statement “A is inversely proportional to B, A =k ”
Trang 3We can express this more succinctly using the composition of functions.2
f−1(f (x)) = x and f (f−1(x)) = x
D e f i n i t i o n
The functions f and g are inverse functions if, for all x in the domain of f ,
g(f (x)) = x and for all x in the domain of g, f (g(x)) = x
EXERCISE 12.1 Verify that the pairs of functions in Example 12.1 actually are inverse functions We’ll do
parts (i) and (ii) below Part (iii) is left as an exercise
i Verify that f (x) = x + 3 and f−1(x) = x − 3 are inverse functions
x→ (x + 3)f f
−1
→ (x + 3) − 3 = x; f−1(f (x)) = f−1(x + 3) = (x + 3) − 3 = x
xf→ (x − 3)−1 → (x − 3) + 3 = x;f f (f−1(x)) = f (x − 3) = (x − 3) + 3 = x
ii Verify that g(x) = 2x and g−1(x) = x2 are inverse functions
x→ 2xg g→−1 2x
2 = x; g−1(g(x)) = g−1(2x) =2x
2 = x
xg→−1 x 2
g
→ 2 x 2
= x; g(g−1(x)) = g x
2
= 2 x 2
= x
D e f i n i t i o n
A function is said to be invertible if it has an inverse function.
Let’s review the characteristic of a function that makes it invertible, able to be undone A function is an input/output relationship such that each input corresponds to a single output
To find the inverse of a function requires that we are able to begin with an output and trace
it back to the corresponding input Therefore no two inputs may share the same output In other words, the function must be 1-to-1; there must be a 1-to-1 correspondence between inputs and outputs
To illustrate this we’ll reconstruct the soda machine example from Chapter 1 Both machine A and machine B can be modeled by functions, but only machine A is 1-to-1
2 Recall that f (g(x)) means find g(x) and use g(x) as the input of f In other words, do f to g(x) Therefore f (f −1 (x))
Trang 4Machine A Machine B
Button # Output Button # Output
Soda machine B has six input buttons, but can give outputs of only Coke and Diet Coke; the function that models this machine is not 1-to-1 Knowing that the machine gave an output
of Coke does not allow us to determine precisely which button was pressed On the other hand, soda machine A, with five selection buttons, each one corresponding to a different type of soda, is modeled by a 1-to-1 function To be invertible a function must be 1-to-1 because the inverse of f must map each output of f to the unique corresponding input; conversely, if f is 1-to-1, then f is invertible
Let f be a function modeling machine A f and its inverse function f−1are shown below
Do we care whether a given function is invertible? It depends on the situation For the soda machines above, it’s probably not very important to be able to tell what button was pressed based on the type of soda that came out of the machine It’s merely important that the relationship between button and soda output be a function so that the input uniquely determines the output
On the other hand, recall the bottle calibration problem from the beginning of the course The entire exercise of calibrating a bottle is worthless unless this calibrated bottle can be used for measuring We want to be able to pour water into the bottle and determine its volume based on the height of the water It’s important that the calibration function is invertible The calibration function, C, takes volume as input and assigns height as output For instance, one liter of water might fill the bottle to a height of 10 cm, in which case C(1) = 10 Once calibrated, the bottle can be used as a measuring device precisely because the calibration function is 1-to-1 C−1 turns the calibration procedure around; the height becomes the input and the volume the output Because the function is 1-to-1 we know that
a height of 10 cm corresponds to a volume of 1 liter, C−1(10) = 1
volume −→ heightC height −→ volumeC−1
EXAMPLE 12.2 Let’s consider a particular bottle and its calibration function C From the information about
Cgiven on the left in the following table we can construct a corresponding table for C−1
Trang 5v C(v) = h h C−1(h) = v (in liters) (in cm) ⇒ corresponding table (in cm) (in liters)
The graph of C is given in Figure 12.2(a) below Since the function C−1reverses the input and output of C, we can graph C−1by reversing the coordinates of the points on the graph
of C
height (cm)
volume
(liters)
4
8
12
(.25, 4)
(.75, 9) (.5, 7)
h = C(v)
height
volume
.25 5 75
(4, 25)
(9, 75) (7, 5)
V = C–1(h)
Figure 12.2
The Relationship Between the Graph of a Function
and the Graph of Its Inverse
We have established that a function is invertible if and only if the function is 1-to-1.3How
is this criterion reflected in the graph of y = f (x)? If f is invertible, then each y-value in the range must correspond to exactly one x-value in the domain
f
x
y0
f
x
(i) f is not invertible on [a, b].
There are y-values corresponding
to more than 1 x-value.
(ii) f is invertible on [a, b].
There is exactly 1 x-value corresponding to each y-value.
Figure 12.3
This gives us a graphical criterion for determining if a function is invertible
3 Recall that A if and only if B means that A and B are equivalent statements If a function is invertible, then it is 1-to-1; if a
Trang 6The Horizontal Line Test. A function f is invertible if and only if every horizontal line intersecting the graph of f intersects it in exactly one point The horizontal line test is the reflection of the vertical line test about the line y = x The vertical line test checks that there
is at most one y for every x; that is, there is at most one output for each input The horizontal line test checks that there is at most one x for every y; that is, there is at most one input for each output Together, they check that the relationship is 1-to-1
Consequence: If a continuous function f is invertible, then f must be either always
increasing or always decreasing A function with a turning point is not invertible; it cannot pass the horizontal line test We can ascertain this information by calculating f If f is continuous on an open interval and fis either always positive or always negative, then
f has no turning points and therefore passes the horizontal line test and is invertible; if
f is continuous and fchanges signs, then f has a turning point and therefore fails the horizontal line test and is therefore not invertible.4
f
x
f is not invertible
f
x
f is invertible
Figure 12.4
As illustrated in Example 12.2, the graph of f−1 is obtained by interchanging the coordinates of the points on the graph of f If (a, b) lies on f , then (b, a) lies on the graph
of f−1 This is equivalent to reflecting the graph of y = f (x) over the line y = x as shown
in the examples below The graphs in Figure 12.5 correspond to the functions and inverse functions given in Example 12.1
x
y
y = x
f
f (x) = x + 3
f–1(x) = x – 3
f–1
3 3 –3
–3
(i)
y = x
g
g (x) = 2x
g–1 (x) = x /2
h
y = x
h–1 (x) = x
Figure 12.5
4 Note that it is not sufficient to check only if f ever equals 0 For example, f (x) = x 3 is invertible since it has no turning points, although f ( 0) = 0.
Trang 7P R O B L E M S F O R S E C T I O N 1 2 1
1 (a) Let S be the function that assigns to each living person a social security number
Is S 1-to-1? Is it invertible?
(b) Let C be the counting function that allows a collection of 30 people to be put in six groups of five people each by “counting off” 1 to 6 Is C 1-to-1? Is it invertible? (c) Let A be the altitude function that assigns to each point in the White Mountains its altitude Is A 1-to-1?
2 The identity function I is the function whose input equals its output: I (x) = x If functions f and g have the property that f (g(x)) = I (x) and g(f (x)) = I (x), then
f and g are inverse functions For each function below, find the inverse function g(x) and verify that f (g(x)) = I (x) and g(f (x)) = I (x)
(a) f (x) = 6x − 3 (b) f (x) = (x − 3)3
3 On the same set of axes, sketch the graphs of the following pairs of functions In parts (a) and (b) find an expression for f−1(x) The graphs of f and f−1(x)are mirror images over the line y = x since the roles of input and output are switched to obtain the inverse function In other words, if (1, 5) is a point on the graph of f , then (5, 1) is
a point on the graph of f−1(x)
(a) f (x) = 2x + 1 and f−1(x) (b) f (x) = x2− 2, x > 0 and f−1(x) (c) f (x) = 10xand f−1(x) (d) f (x) = 2−xand f−1(x)
4 Suppose f (v) is a calibration function for a bucket f takes volumes (in liters) as inputs and gives heights (in inches) as outputs Suppose f (1) = 4
(a) What is f−1(4)?
(b) What is the meaning of f−1(4) in physical terms?
(c) Is f−1(4) greater than f−1(1)? Explain in terms of the physical situation
5 Which of the following functions are invertible on the domain given? Explain (a) P (w) is the price of mailing a package weighing w ounces; w ∈ (0, 50]
(b) T (t) is the temperature at the top of the Prudential Center in Boston at time t, t measured in days, where t = 0 is February 1, 1998; t ∈ [0, 365]
(c) C(w) is the cost of w pounds of ground coffee at a particular shop where coffee
is sold by weight at a fixed price per pound; w ∈ [0, 2]
(d) M(t) is the mileage on a car t days after it was purchased; t ∈ [0, 365]
6 Let f (x) = x3+ 3x2+ 6x + 12
(a) Make a convincing argument that f (x) is invertible (It is not adequate to say
it looks 1-to-1 on a calculator How can you be absolutely sure it is 1-to-1 on (−∞, ∞)?)
(b) Find three points that lie on the graph of f−1(x) (Approximations are not ade-quate.) Explain your reasoning
Trang 87 For each of the functions graphed below, determine whether or not the function is invertible If it is not, restrict the domain to make it invertible Then sketch f−1, labeling any asymptotes and labeling two points on the graph of f−1
x =
x
(2, 4)
1
f (x)
f (x)
–π
( /2
–π/2 x =π/2 , –1)
π
( /2, 1)
π
( /4, 1)
8 The graphs of f (t), g(t), and h(t) are given below
y
t
g (t)
–π
h (t)
y
t
x = x =
2
–π
2
π
2
y
t
f (t)
π –π
–π
2
π
2
π
2
True or False?
(a) The function f (t), restricted to the domain [0, π ], is invertible
(b) The function g(t), restricted to the domain [0, π ], is invertible
(c) The function h(t), restricted to the domain (−π2,π2), is invertible
(d) The function f (t), restricted to the domain [−π
2,π
2], is invertible
9 Which of the following functions are invertible?
(a) The function that assigns to each current senator the state he or she represents (b) The function T (t) that gives the temperature in Moab at time t
(c) The function C(d), whose domain is the set of all performances of Broadway’s A Chorus Line, and whose output is the cumulative number of people who have seen this show on Broadway
(d) The function L(d), whose domain is the set of all performances of Broadway’s
The Lion King, and whose output is the number of people seeing this Broadway show on the designated date
Trang 912.2 FINDING THE INVERSE OF A FUNCTION
If an invertible function is given by a table of values, the inverse function is constructed
by interchanging the input and output columns If an invertible function is presented graphically, the graph of f−1is obtained by reflecting the graph of f over the line y = x Suppose a function is given analytically The subject of this section is how to arrive at an expression for its inverse function
EXAMPLE 12.3 Suppose f is the function that doubles its input and then adds 3: f (x) = 2x + 3 What is
its inverse function?
SOLUTION To undo the function f , do we subtract 3 and then divide by 2, or do we first divide by 2
and then subtract 3; is f−1(x) =x−32 or is f−1(x) =x2− 3? We could try each out, since
we know that f−1(f (x))should be x, or we can think about it in the following way
To undo the process, you must first remove your shoes, then your socks The last thing you did is the first thing you undo
Accordingly, to undo f (x), we first subtract 3 and then divide the result by 2: f−1(x) =
x−3
2
x
multiply by 2
add 3
2x
2x + 3
divide by 2
subtract 3
f–1(2x + 3) = x
f (x) = 2 x + 3
Check:
f−1(f (x)) = f−1(2x + 3) = (2x+3)−32 =2x2 = x
f (f−1(x)) = f x−32 = 2x−32 + 3 = (x − 3) + 3 = x The three functions from Example 12.1 and the function from the example above had simple enough formulas that we could guess how to “undo” them in order to find formulas for their inverses As this is not always the case, we need an analytic method for finding a formula for the inverse function
EXAMPLE 12.4 Let f (x) =x+1
x−2 Find a formula for f−1(x)
SOLUTION In this example it is not very easy to figure out how to undo the action of f , so we’ll use a
different approach: f−1reverses the output and input of f Let’s denote the output of f by
yand the input by x f is given by y =x+1x−2 We interchange the input and output (x and y)
to obtain f−1
Trang 10x = y + 1
y − 2, where x is the input of f−1and y is the output of f−1 We solve for y to get the output of
f−1in terms of the input
x =y + 1
y − 2 Get y out of the denominator
by multiplying both sides by (y − 2)
x(y − 2) = y + 1 Multiply out
xy − 2x = y + 1 Gather the y’s together
xy − y = 2x + 1 Factor out y
y(x − 1) = 2x + 1 Solve for y
y =2x + 1
x − 1
So f−1(x) = 2x+1x−1
To summarize, we find a formula for f−1by exploiting the fact that the inverse of f reverses the roles of input and output
Write y = f (x), and then interchange the variables x and y
Solve for y in terms of x We’ll obtain y = f−1(x)
EXAMPLE 12.5 Let f (x) = 4x3
+ 2 Find f−1(x)if f is invertible
SOLUTION We know that f is invertible because f(x) = 12x2is positive for all x = 0 This indicates
that f is a cubic polynomial with no turning points, so it’s 1-to-1 Set y = 4x3+ 2 and then interchange the roles of x and y to find the inverse relationship
x = 4y3+ 2 4y3= x − 2
y3=x − 24
y = x − 2
4
1/3
f−1(x) = x − 2
4
1/3