It is not possible for a single element a to have two
different inverses b 1 and b 2 . This is established by noting that e = a*b 2 so that b 1 = b 1 *e = b 1 *(a*b 2 ) =
(b 1 *a)*b 2 = e*b 2 = b 2 .
From the symmetry of the definition, we have that if
b is the inverse of a, then a is also the inverse of b. Consequently, the inverse of an inverse is the original element. Phrased in terms of addition, this reads –(–a) = a
and in terms of multiplication as:
The inverse of an element a is often denoted a –1 , especially 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 f with domain D and range R, f : D → R,
is said to be invertible or to have an inverse function if,
for each possible output y of 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 x that produces 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 output 11 is produced from the input of 4, and so we have
f
–1 (11) = 4. In general, an output of y is produced
from an input x =
, and so f
–1 (y) =
. (This
formula is obtained by solving for x in the equation:
y = 2x + 3 to yield x =
.)
Since f
–1 (y) is the input that produces the output y,
and x is the input that produces the output f(x), the following relations hold:
f(f
–1 (y)) = y for all values y in the range of f
and
f
–1 (f(x)) = x for all x in the domain of f
This explains the awkward notation for the inverse
function: In the study of the COMPOSITION of functions,
f m denotes the composite f 0 f 0 … 0 f (m times), and we
have f
m
0 f
n = f
m+n . To give meaning to the quantity f
0 ,
this rule states that f 0 f
0 = f
1
0 f
0 = f
1+0 = f, suggesting
that we should set f
0 (x) = x for all values x. Consequently, the statement f
–1
0 f
1 = f
0 suggests that f
–1 (f(x))
= x for 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 x and the output as the variable y. This
can lead to some confusion. For instance, to compute
the inverse function of y = f(x) = x
3 + 2 we solve for the
input x in the equation in terms of the output y to yield,
x =
, but we interchange the x and y variables so
that x denotes the new input and y the 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 equation 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 = x returns the y-axis
to the vertical position and the x-axis to the horizontal
position, but also flips the curve drawn across the diagonal 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 squaring function y = x
2 : 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 function, 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 function. One defines the INVERSE TRIGONOMETRIC FUNCTIONS, 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
1
1
a
= a
280 inverse function
different inverses b 1 and b 2 . This is established by noting that e = a*b 2 so that b 1 = b 1 *e = b 1 *(a*b 2 ) =
(b 1 *a)*b 2 = e*b 2 = b 2 .
From the symmetry of the definition, we have that if
b is the inverse of a, then a is also the inverse of b. Consequently, the inverse of an inverse is the original element. Phrased in terms of addition, this reads –(–a) = a
and in terms of multiplication as:
The inverse of an element a is often denoted a –1 , especially 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 f with domain D and range R, f : D → R,
is said to be invertible or to have an inverse function if,
for each possible output y of 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 x that produces 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 output 11 is produced from the input of 4, and so we have
f
–1 (11) = 4. In general, an output of y is produced
from an input x =
, and so f
–1 (y) =
. (This
formula is obtained by solving for x in the equation:
y = 2x + 3 to yield x =
.)
Since f
–1 (y) is the input that produces the output y,
and x is the input that produces the output f(x), the following relations hold:
f(f
–1 (y)) = y for all values y in the range of f
and
f
–1 (f(x)) = x for all x in the domain of f
This explains the awkward notation for the inverse
function: In the study of the COMPOSITION of functions,
f m denotes the composite f 0 f 0 … 0 f (m times), and we
have f
m
0 f
n = f
m+n . To give meaning to the quantity f
0 ,
this rule states that f 0 f
0 = f
1
0 f
0 = f
1+0 = f, suggesting
that we should set f
0 (x) = x for all values x. Consequently, the statement f
–1
0 f
1 = f
0 suggests that f
–1 (f(x))
= x for 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 x and the output as the variable y. This
can lead to some confusion. For instance, to compute
the inverse function of y = f(x) = x
3 + 2 we solve for the
input x in the equation in terms of the output y to yield,
x =
, but we interchange the x and y variables so
that x denotes the new input and y the 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 equation 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 = x returns the y-axis
to the vertical position and the x-axis to the horizontal
position, but also flips the curve drawn across the diagonal 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 squaring function y = x
2 : 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 function, 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 function. One defines the INVERSE TRIGONOMETRIC FUNCTIONS, 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
1
1
a
= a
280 inverse function