2.7 INVERSE FUNCTIONS AND PARAMETRIC EQUATIONS

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Chapter 2 Functions

2.7 I N V E R S E F U N C T I O N S A N D P A R A M E T R I C E Q U A T I O N S

Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable.

R. W. Hamming

B[ efore I zeroed in on

math], I found the whole university just fascinating. There was that rare books library that had everything in the world in it. I loved the history courses. I loved the English courses. I enjoyed physics very much. Even in my senior year I took courses all over the map--almost as much philosophy as mathematics, almost as much history as mathematics, almost as much English as mathematics, almost as much Spanish as mathematics.

Mary Ellen Rudin

In Section 2.2 we noted that a function can be considered as a set of ordered pairs in which no two different pairs have the same first component. For each first number, any ordered pair can have exactly one second number.

As an example, suppose function f is f x 2x, where the domain is D 2, 1, 0, 1, 2. In terms of ordered pairs, f may be written as

f 2, 4, 1, 2, 0, 0, 1, 2, 2, 4.

The range R of f is given by R 4, 2, 0, 2, 4. Now suppose we interchange the two entries in each of the ordered pairs of f and get a new set of ordered pairs that we denote by g.

g 4, 2, 2, 1, 0, 0, 2, 1, 4, 2

We can make several observations concerning g:

1. Since no two pairs of g have the same first numbers, g is a function.

2. The domain of g is D 4, 2, 0, 2, 4 and the range is

R 2, 1, 0, 1, 2.

3.

Function

g

is

gx

x ,

where

x

D.

2

4. The domains and ranges of f and g are interchanged; D R and

R D.

Let us consider the composition function g f defined by g f x g f x for x in D. For instance,

when x is 2, g f 2 g4 2

when x is 1, g f 1 g2 1

and so on. In general,

g f x g2x 2x x. 2

We get a similar result for f g:

f gx f gx f

x 2

2 x 2

x.

The functions f and g that we have been discussing are related in a special way-- one is the inverse of the other. Since f gx x and g f x x, we may say that each of the functions "undoes" or neutralizes the other. If we start with x, apply f and get f x, and then apply g to f x, we get back to x.

Schematically, think of a function as a map that sends each element in the domain to a corresponding range element. The inverse function sends each element of the range to the original element of the domain. A diagram like Figure 43 may help clarify the relationship. Observe that the diagram may be read in either direction so that applying f and then g, or g and then f , always yields the initial input.

2.7 Inverse Functions and Parametric Equations

f

x

f (x)

g( f (x))

Domain of f g

119

FIGURE 43 Inverse function: g f x x for every x in the domain of f.

Definition: inverse functions

Suppose f and g are functions that satisfy two conditions: g f x x for every x in the domain of f , and f gx x for every x in the domain of g. Then f and g are inverses of each other.

Characterization of Inverse Functions

Suppose f is a function described as a set of ordered pairs such that no two pairs have the same second element, f x, y y f x.

Let g be the set of ordered pairs obtained by interchanging the elements of each pair of f. If g is a function, then f and g are inverses of each other.

Notation for Inverse Functions

Suppose g is the inverse of function f. It is customary to denote g by f1. Note that

f

1

does

not

mean

the

reciprocal

of

f

,

which

we

would

write

as

1 f

.

Replacing

g

by

f1 in the above definition gives an important pair of identities.

Inverse function identities

f 1 f x x for each x in the domain of f. f f 1x x for each x in the domain of f 1.

EXAMPLE 1 Verify inverse

Strategy: Simply verify that f f 1x x and f 1 f x x for every real

number x.

(a) Verify that

f x 2x 3 and

are inverses of each other. (b) Draw graphs of y f x and y f1x.

f 1x x 3 2

y

Solution

4 3 (0, 3)

(a) Follow the strategy. For every real number x,

2 y = f (x) 1

y=x (3, 0)

f 1 f x f 12x 3 2x 3 3 2x x.

2

2

? 4 ? 3 ? 2 ? 1? 1 ?2

x 1234

y = f ?1(x)

f f 1x f x 3 2 x 3 3 x 3 3 x.

2

2

?3 (? 3, ? 3 )

?4 FIGURE 44

Therefore, the given functions are inverses of each other.

(b) The graphs of y 2x 3 and y x 3 are the lines shown in 2 Figure 44.

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Chapter 2 Functions

Graphs of Inverse Functions

The graphs of the functions y f x and y f 1x in Figure 44 appear symmet-

ric with respect to the line y x, that is, each graph is a reflection of the other

through that line. We want to show that for any pair of inverse functions, the graphs

y

of f and f 1 are symmetric about the line y x.

Suppose that a, b is any ordered pair in f. This implies that b, a is an ordered

pair in f 1. If we denote the line y x by L, then we must show that the points

Q(b, a) L: y = x

M P(a, b)

Pa, b and Qb, a are reflections of each other in L; that they are equidistant from

L and that the line through P and Q is perpendicular to L. The midpoint M of

ab ab

segment PQ has coordinates

,

. Direct computation verifies that M

2

2

lies on line L and that P and Q are equidistant from L. Further, the line through P

x

and

Q

has

slope

given

by

m

b a

a b

1,

and

hence

is

perpendicular

to

L

since

FIGURE 45

the slope of L is 1. See Figure 45.

Graphs of inverse functions

For each point a, b on the graph of y f x, the point b, a belongs to the graph of y f 1x; that is, coordinates of every point are interchanged.

The graph of y f 1x is a reflection through the line y x of the graph of y f x.

The domain of f becomes the range of f 1, and conversely; that is, the

domains and ranges are interchanged.

Finding Equations for Inverse Functions

It is not always easy to find an equation that describes the inverse of a particular function. In many cases, however, the inverse function does have an equation that can be found readily by a straightforward algorithm. The key is the observation that finding f 1 from f requires interchanging x and y in each ordered pair x, y of f to obtain the ordered pairs in f 1.

Algorithm: finding an equation for an inverse function

Step 1. Write the equation defining f in the form y f x. Step 2. Interchange y and x to get x f y. Step 3. Solve the equation x f y for y, and adjust the domain as needed. Step 4. The result is y f 1x.

Strategy: First write y 2x 1 and then follow the steps of the algorithm.

EXAMPLE 2 Formula for inverse Find a formula for the inverse of f x 2x 1 and verify that f f 1x x.

Solution Follow the strategy.

Step 1 Step 2

Step 3

y 2x 1

x 2y 1

y

x

1 ;

therefore,

f 1x

x

1 .

2

2

2.7 Inverse Functions and Parametric Equations

121

To verify that f f 1x x,

f f 1x f x 1 2 x 1 1 x 1 1 x.

2

2

EXAMPLE 3 A function and its inverse Let gx 3 x. Find the domain and range of g, and find a formula for the inverse function g1, and state its domain and range. Sketch a graph of both g and g1.

Solution

By the domain convention, g is defined when 3 x 0, so the domain of g is the interval (, 3, and since a square root is always nonnegative, the range is the interval 0, ). To find a formula for g1, we follow the steps of the algorithm.

Steps 1 and 2. Write y 3 x, and interchange variables, x 3 y.

Step 3. To solve for y, we begin by squaring, x 2 3 y. Then y 3 x 2. For the inverse function, the domain and range of f are interchanged, so the domain is the interval 0, ), and the range is , 3.

Step 4. With the restricted domain, we have g1x 3 x 2, x 0.

The graph of g is the upper half of a parabola opening to the left; the graph of g1 is the right half of a parabola opening downward. See Figure 46.

y g(x) = ? x + 3

(0, 3)

(? 1, 2)

(2, 1)

x (3, 0)

y

(0, 3) g? 1(x) = 3 ? x2, x 0 (1, 2)

( 3, 0) x

(2, ? 1)

(a)

(b)

FIGURE 46

Using Parametric Mode to Graph Inverses

Graphing calculators have a mode of graphing called parametric mode which

allows us to visualize directly what happens when we interchange the ordered pairs

in a function. In parametric mode we use separate equations for the x- and y-coor-

dinates, defining both coordinates as functions of a new variable. We study para-

metric equations in some detail later in the book, but we make use of the parametric

mode on graphing calculator whenever it can help us to understand ideas. To illustrate, suppose we have a pair of equations, x t, y t 2. Then, for each

value of t, we get a pair of numbers x, y with the property that y is the square of x. For a given set of t-values, if we graph the pairs x, y thus determined, we get a set of points satisfying the equation y x 2. That is, the two equations x t, y t 2, together define part (or all) of the parabola we have previously defined by the single equation y x 2. The variable t is called a parameter, and the equations x t, y t 2, are called parametric equations for the parabola y x 2.

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Chapter 2 Functions

HISTORICAL NOTE INVERSE FUNCTIONS AND CRYPTOGRAPHY

Encoding and decoding secret

Even though the coding functions

messages depends on functions and

have become extremely complex,

their inverses. Each letter is assigned

involving continual modifications,

a number (often, its place in the

until very recently all cryptology

alphabet) and a coding function is

algorithms required the same work

applied to the number. A simple

of the cryptographer and the

Caesar cipher is given by

decrypter. Knowing how to encode

f n n 5 (mod 26), meaning

(which required the coding function)

that n 5 is reduced by multiples

meant knowing how to decode

of 26 when necessary. S 19

(which required the inverse function).

becomes f 19 24 X. SEND

All this has changed with the

MONEY becomes XJSI RTSJD.

Decoding uses the inverse function f 1n n 5 (mod 26).

In a slightly more complex function, Fn 3n 5 (mod 26),

This Renaissance crypt-analysis instrument works by rotating the inner disk against the stationary

outer disk.

invention of trapdoor codes, which have efficient algorithms for both functions and inverses, but for which inverses are effectively impossible to discover, so no one can break the

the letter S, which corresponds to 19,

code.

becomes

The most impressive are the RST codes (named

F19 3 ? 19 5 mod 26 62 (mod 26) 10.

Therefore F (19) 10, and since J corresponds to

for their discoverers). These depend on finding large primes whose products cannot easily be factored. A few minutes of computer time can produce 100-digit primes, but factoring the

10, S becomes J. The inverse function to decode JTUQ RXUTB is given by F1n 9n 7

product of two such numbers would typically require millions of years.

(mod 26).

The set of points satisfying the equation x y 2 cannot be graphed as a single function on a graphing calculator, but we could graph two functions, y x and y x. Together, they form a parabola opening to the right, the graph of the inverse relation of the function f x x 2. In parametric mode, however, it is just as easy to use the equations y t, x t 2 for the parabola x y 2 as it is to use x t, y t 2 for the parabola y x 2. In fact, for any function y f x, we can use the parametric equations x t, y f t for the function, and the parametric equations y t, x f t define the inverse relation (which may be a function, but need not be). The fact that there can be different sets of parametric equations to define the same curve will not concern us here; we have a method to allow us to write a set of parametric equations for any given function and for its inverse.

Parametric equations for a function and its inverse

To graph a function y f x in parametric form, use equations

X T, Y f(T),

and to graph the inverse of f (which need not be a function), use equations

Y T, X fT.

2.7 Inverse Functions and Parametric Equations

123

We discuss below conditions that will allow us to tell when the inverse of f is a function.

TECHNOLOGY TIP Parametric graphing and t-range

When graphing in parametric mode, we must set a t-range as well as x-and y-ranges. If we are using equations

X T, Y f(T),

we can see all of the graph that appears in the window if we set the t-range to match the x-range. The tStep (or , t-Pitch or ) Step determines how many points will be plotted. Usually something around 0.1 gives a reasonable graph without taking too long. Experiment with your calculator.

Limiting the t-range allows us to graph just a portion of f. For the simple example x t, y t 2, setting a t-range from 1 to 1 gives the "tip" of the parabola, from the point (1, 1) to the point (1, 1). Similarly, for the t-range 0, 5 in the decimal window, we see only the part of the parabola in the first quadrant. Again, experiment.

y = g(x)

y = g? 1(x)

[? 4.5, 4.5] by [? 3.1, 3.1] FIGURE 47

EXAMPLE 4 Parametric graphs Let gx 3 x. In parametric mode sketch a calculator graph of both g and g1 on the same screen.

Solution This is the same function as in Example 3.

With the calculator in parametric mode (make sure you know how to set the proper mode on your calculator), enter

X T, Y (3 T),

and to graph the inverse of g use equations

Y T, X (3 T).

For the t-range, use the same interval as the x-range. If, for example you are using a decimal window with Xmin 4.7, Xmax 4.7, use the same values for Tmin and Tmax and set Tstep .1. When you graph, both g and g1 should appear on the same screen, as in Figure 47. Note that when you TRACE, you can read the coordinates of all three variables, x, y, and t.

Existence of Inverse Functions

Not every function has an inverse that is a function. For the function yx 2, if we interchange x and y to get x y 2 and then solve for y, we have y x. For each x 0 there are two corresponding values of y, so we do not have a function.

When we say that a function "has an inverse," we mean that its inverse is a function.

To determine whether or not a function f has an inverse, we look back at ordered pairs x, y of f , where y f x. We know that for every x in the domain D of f , there is exactly one value of y. Now suppose we interchange x and y. The set of ordered pairs y, x will be a function only if for each y there is exactly one x. Therefore, for y f x to have an inverse each x must correspond to exactly one y so f is a function and every y to exactly one x. We call such a function a one?one function.

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Chapter 2 Functions

Existence of an inverse function--Part I

A function has an inverse if and only if the function is one?one.

How can we tell when a function is one?one? The best way is to draw a graph. Section 2.2 introduced the vertical line test to determine whether a graph represents a function. If we combine this test with the horizontal line test described below, we can determine whether or not a graph is that of a one?one function.

Horizontal line test

If every horizontal line intersects the graph of a function in at most one point, then that function is one?one. Therefore, it has an inverse.

Figure 48 shows graphs of three functions. The graphs in panels (a) and (b) represent one?one functions while that in panel (c) does not. In panel (c) horizontal lines like L1 intersect the graph at one point, but lines such as L2 intersect the graph at more than one point.

y

y

y

L1

L L2

L

x

x

x

(a)

(b)

(c)

FIGURE 48

A useful criterion for determining whether the inverse of a function is itself a function comes from a property we can read from a graph. A function whose graph rises as we move from left to right is called an increasing function. Similarly, if the graph drops (goes down) as we move from left to right, the function is called decreasing. We give a formal definition, but visualizing the distinction as in Figure 49 is at least as helpful.

y

y

Increasing (a)

Decreasing

x

x

(b) FIGURE 49

2.7 Inverse Functions and Parametric Equations

125

Definition: Increasing functions, decreasing functions

Suppose b and c are any numbers in the domain of a function f where b c. The function f is an increasing function if f b f c, and it is a decreasing function if f b f c.

In terms of increasing and decreasing functions we can restate the condition for the existence of an inverse function.

Existence of an inverse function--Part II

Function f has an inverse if f is either an increasing function or a decreasing function.

Strategy: In each case sketch a graph and then see if every horizontal line intersects the graph in at most one point.

EXAMPLE 5 Existence of inverse Determine which functions have inverses:

(a) f x 2x Solution

(b)

gx

1 x

(c) hx x 2 2x.

(a) The graph of y 2x is a line as shown in Figure 50(a). Every horizontal line

intersects the graph at exactly one point, so f is a one?one function and it has

an inverse.

(b)

The

graph

of

y

1 x

is

shown

in

Figure

50(b).

Every

horizontal

line

except

y 0 (the x-axis) intersects the graph at one point and the line y 0 does not

intersect at any point. Thus g is a one?one function and it has an inverse.

(c) The graph of y x 2 2x is a parabola, as shown in Figure 50(c). Clearly

there are horizontal lines that intersect the graph in more than one point, so h

does not have an inverse.

y

y

y

y = 2x

y = x2 ? 2x

y

=

1 x

x

x

x

(a)

(b)

(c)

Strategy: In each case draw a graph and determine whether the function is increasing, decreasing, or neither.

FIGURE 50

EXAMPLE 6 Increasing and decreasing functions Determine whether the given function is increasing, decreasing, or neither.

(a) f x x 3 (b) gx x 2 (c) hx x 2, for x 0

Solution Follow the strategy.

(a) The graph of y x 3 is shown in Figure 51a. Clearly f is an increasing function.

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