280 inverse function

It is not possible for a single element ato have two

different inverses b1and b2. This is established by not-

ing that e= a*b2so that b1= b1*e= b1*(a*b2) =

(b1*a)*b2= e*b2= b2.

From the symmetry of the definition, we have that if

bis the inverse of a, then ais also the inverse of b. Con-

sequently, the inverse of an inverse is the original ele-

ment. Phrased in terms of addition, this reads –(–a) = a

and in terms of multiplication as:

The inverse of an element ais often denoted a–1, espe-

cially if the binary operation under consideration can be

interpreted as a type of multiplication. For example, the

inverse of a square

MATRIX

A, if it exists, is denoted A–1.

inverse function (inverse mapping, reverse function)

A

FUNCTION

fwith domain Dand range R, f : D →R,

is said to be invertible or to have an inverse function if,

for each possible output yof the function, y∈R, there

is one, and only one, input x∈D, that produces that

output. We write x= f–1(y) for the input xthat pro-

duces the given output y. (Thus x= f–1(y) if, and only

if, f(x) = y.) This then defines a function f–1: R→D,

called the inverse function to f. In some sense, the

inverse function “undoes” the original function.

For example, consider the function on real numbers

that doubles an input and adds 3: f(x) = 2x+ 3. The out-

put 11 is produced from the input of 4, and so we have

f–1(11) = 4. In general, an output of yis produced

from an input x= , and so f–1(y) = . (This

formula is obtained by solving for xin the equation:

y= 2x+ 3 to yield x= .)

Since f–1(y) is the input that produces the output y,

and xis the input that produces the output f(x),the fol-

lowing relations hold:

f(f–1(y)) = yfor all values yin the range of f

and

f–1(f(x)) = xfor all xin the domain of f

This explains the awkward notation for the inverse

function: In the study of the

COMPOSITION

of functions,

fmdenotes the composite f0f0…0f(mtimes), and we

have fm0fn= fm+n. To give meaning to the quantity f0,

this rule states that f0f0= f10f0= f1+0 = f, suggesting

that we should set f0(x) = xfor all values x. Conse-

quently, the statement f–10f1= f0suggests that f–1(f(x))

= xfor all x, indicating that f–1 is the appropriate

notation for the inverse function. The superscript of

–1 should not be confused with the operation of

inversion. (We write (f(x))–1 to denote , and leave

f–1(x) to mean the inverse function of f.)

It is customary to denote the input of a real function

as the variable xand the output as the variable y. This

can lead to some confusion. For instance, to compute

the inverse function of y= f(x) = x3+ 2 we solve for the

input xin the equation in terms of the output yto yield,

x= , but we interchange the xand yvariables so

that xdenotes the new input and ythe new output: y=

. This yields the formula f–1(x) = for the

inverse function.

As the formulae y= f(x) and x= f–1(y) represent

exactly the same equation, the two formulae yield

exactly the same curves when plotted against a pair of

x- y-coordinate axes. Following the convention to

interchange the x- and y-variables for the second equa-

tion to write y= f–1(x) is tantamount to interchanging

the x- and y-axes in the graph of the curve. Flipping the

graph across the diagonal line y= xreturns the y-axis

to the vertical position and the x-axis to the horizontal

position, but also flips the curve drawn across the diag-

onal line. Thus the graphs of y= f(x) and y= f–1(x) are

mirror images of each other across a diagonal line.

Not every function possesses an inverse function.

For example, there is no inverse function to the squar-

ing function y= x2: some outputs arise from more than

one possible input. (The output of 4, for instance,

arises from the two inputs 2 and –2.) However, it is

often possible to restrict a function to a certain portion

of its domain and define an inverse function for that

restricted domain. For instance, for the squaring func-

tion, if we require that only nonnegative inputs are to

be considered, then an inverse function does exist: we

have y= √

–

x(the positive square root) as inverse func-

tion. One defines the

INVERSE TRIGONOMETRIC FUNC

-

TIONS

, for example, by restricting to a suitable portion

of the domain.

If the function y= f(x),then the derivative of the

inverse function y= f–1(x) is given by:

3

√x– 2

3

√x– 2

3

√y– 2

1

––

f(x)

y– 3

–––

2

y– 3

–––

2

y– 3

–––

2