Panorama histórico del Museo Regional de Chiapas

In our introductory example, we saw that f (P )= 30p−1/3_{and g(D)= 27 000D}−3_were

inverse functions. Because of the concrete interpretation of the symbols P and D, it was natural to describe the functions the way we did. In other circumstances, it may be convenient to use the same variable as argument in both f and g. In Example 1(a), we saw that f (x)= 4x − 3 and g(y) = 1₄y+3₄were inverses of each other. If also we use x instead of yas the variable of the function g, we find that

f (x)= 4x − 3 and g(x)= 1₄x+3₄ are inverses of each other (∗) In the same way, on the basis of Example 1(b) we can say that

f (x)= (x + 1)1/5 and g(x)= x5− 1 are inverses of each other (∗∗) There is an interesting geometric property of the graphs of inverse functions. For the pairs of inverse functions in (∗) and (∗∗), the graphs of f and g are mirror images of each other with respect to the line y= x. This is illustrated in Figs. 3 and 4.

g(x) ! 1 4x # 34 f_{(x) ! 4x " 3 y ! x} y x 2 "2 4 6 8 2 "2 4 6 g(x) ! x5 _{" 1} f(x) ! (x # 1) y ! x 3 1 "2 "2 1 y x 1 5 3

Figure 3 fand g are inverses of each other Figure 4 fand g are inverses of each other

Suppose in general that f and g are inverses of each other. The fact that (a, b) lies on the graph f means that b= f (a). According to (1), this implies that g(b) = a, so that (b, a) lies on the graph of g. Because (a, b) and (b, a) lie symmetrically about the line y= x (see Problem 8), we have the following conclusion:

When two functions f and g are inverses of each other, then the graphs of y = f (x) and y = g(x) are symmetric about the line y = x. (The units on the coordinate axes must be same.)

(3)

NOTE 2 When the functions f and g are inverses of each other, then by definition (1), the

equations y= f (x) and x = g(y) are equivalent. The two functions actually have exactly the same graph, though in the second case we should think of x depending on y, instead of the other way around. On the other hand, the graphs of y = f (x) and y = g(x) are symmetric about the line y= x.

For instance, Examples 4.5.3 and 5.1.3 discuss demand and supply curves. These can be thought of as the graphs of a function where quantity Q depends on price P , or equivalently of the inverse function where price P depends on quantity Q.

In all the examples examined so far, the inverse could be expressed in terms of known formulas. It turns out that even if a function has an inverse, it may be impossible to express it in terms of a function we know. Inverse functions are actually an important source of new functions. A typical case arises in connection with the exponential function. In Section 4.9 we showed that y= ex _{is strictly increasing and that it tends to 0 as x tends−∞ and to ∞}

as x tends to∞. For each positive y there exists a uniquely determined x such that ex _{= y.}

In Section 4.10 we called the new function the natural logarithm function, ln, and we have the equivalence y = ex _{⇐⇒ x = ln y. The functions f (x) = e}x _{and g(y)= ln y are}

therefore inverses of each other. Because the ln function appears in so many connections, it is tabulated, and moreover represented by a separate key on many calculators.

If a calculator has a certain function f represented by a key, then it will usually have another which represents its inverse function f−1_{. If, for example, it has an e}x-key,

it also has an ln x-key. Since f−1(f (x)) = x, if we enter a number x, press the

f-key and then press the f−1-key, then we should get x back again. Try to enter 5, use

the ex-key and then the ln x-key. You should then get 5 back again. (One reason why you

might not get exactly 5 is a rounding error.)

If f and g are inverses of each other, the domain of f is equal to the range of g, and vice versa. Consider the following examples.

E X A M P L E 2 The function f (x) =√3x+ 9, defined in the interval [−3, ∞), is strictly increasing and hence has an inverse. Find a formula for the inverse. Use x as the free variable for both functions.

Solution: When x increases from−3 to ∞, f (x) increases from 0 to ∞, so the range of f is [0,∞). Hence f has an inverse g defined on [0, ∞). To find a formula for the inverse, we solve the equation y=√3x+ 9 for x. Squaring gives y2_{= 3x + 9, which solved for x}

gives x=1

3y2− 3. Interchanging x and y in this expression to make x the free variable, we

find that the inverse function of f is y= g(x) =1

S E C T I O N 5 . 3 / I N V E R S E F U N C T I O N S 141 "3 3 3 "3 f(x) ! !3x # 9 g(x) ! 1 3x2 " 3 y ! x y x Figure 5

E X A M P L E 3 Consider the function f defined by the formula f (x)= 4 ln(√x+ 4 − 2). (a) For which values of x is f (x) defined? Determine the range of f . (b) Find a formula for its inverse. Use x as the free variable.

Solution:

(a) In order for√x+ 4 to be defined, x must be ≥ −4. But we also have to make sure that√x+ 4 − 2 > 0, otherwise the logarithm is not defined. Now,√x+ 4 − 2 > 0 means that√x+ 4 > 2, or x + 4 > 4, that is, x > 0. The domain of f is therefore (0,∞). As x varies from near 0 to ∞, f (x) increases from −∞ to ∞. The range of f is therefore (−∞, ∞).

(b) If y = 4 ln(√x+ 4 − 2), then ln(√x+ 4 − 2) = y/4, so that√x+ 4 − 2 = ey/4 and then√x+ 4 = 2 + ey/4. By squaring each side we obtain x+ 4 = (2 + ey/4)2= 4+ 4ey/4_{+ e}y/2_{, so that x = 4e}y/4_{+ e}y/2_{. The inverse function, with x as the free}

variable, is therefore y= ex/2_{+ 4e}x/4_{. It is defined in (−∞, ∞) with range (0, ∞).}

P R O B L E M S F O R S E C T I O N 5 . 3

1. Demand D as a function of price P is given by D=32

5 − 3 10P Solve the equation for P and find the inverse function.

2. The demand D for sugar in the US in the period 1915–1929, as a function of the price P , was estimated by H. Schultz as

D= f (P ) = 157.8

P0.3 (D and P are measured in appropriate units) Solve the equation for P and so find the inverse of f .

3. Find the domains, ranges, and inverses of the functions given by the four formulas

⊂

⊃

x −4 −3 −2 −1 0 1 2

f (x) −4 −2 0 2 4 6 8

(a) Denote the inverse of f by f−1_{. What is its domain? What is the value of f}−1_(2)? (b) Find a formula for a function f (x), defined for all real x, which agrees with this table. What

is the formula for its inverse?

5. Why does f (x)_{= x}2_{, for x in (−∞, ∞), have no inverse function? Show that f restricted to} [0,_{∞) has an inverse, and find that inverse.}

6. Formalize the following statements:

(a) Halving and doubling are inverse operations.

(b) The operation of multiplying a number by 3 and then subtracting 2 is the inverse of the operation of adding 2 to the number and then dividing by 3.

(c) The operation of subtracting 32 from a number and then multiplying the result by 5/9 is the inverse of the operation of multiplying a number by 9/5 and then adding 32. “Fahrenheit to Celsius, and Celsius to Fahrenheit”. (See Example 1.6.4.)

7. If f is the function that tells you how many kilograms of carrots you can buy for a specified amount of money, then what does f−1_{tell you?}

8. (a) Draw a coordinate system in the plane. Show that points (3, 1) and (1, 3) are symmetric about the line y_{= x, and the same for (5, 3) and (3, 5).}

(b) Use properties of congruent triangles to prove that points (a, b) and (b, a) in the plane are symmetric about the line y_{= x. What is the point half-way between these two points?}

⊂

⊃

(a) f (x)_{= (x}3_{− 1)}1/3 _{(b) f (x)=}x+ 1

x− 2 (c) f (x)= (1 − x 3₎1/5_{+ 2}