# Inverse Functions

Short Answer: If f (x) = x, its inverse (under the
operation of function composition)

is just itself, i.e. . Why? Because
and

. A more substantive example is better, such
as the function f : (0,∞) defined

by f (x) = x^{2}. Then because
and

for all x in (0,∞).

Long Answer: To understand inverses we must first understand "identitites." For
any

algebraic operation , the identity is that which leaves things unchanged under
the operation.

For example, under addition the ** right identit**y is 0 because a + 0 = a for any
number

a. That is, adding 0 to a on the right doesn't change a . Similarly, for
multiplication, 1 is

the right identity, because multiplying any number on the right by 1 leaves that
number

unchanged; i.e. a ·1 = a. Sometimes we can find a** left identity** too. For example,
the left

identity under addition is also 0 because 0 + a = 0 for any a. So 0 is both a
right and left

identity for addition. Similarly, 1 is both a right and left identity for
multiplication. When

it works on both sides it's called a two -sided identity, or just the **identity**.

Extend this idea now to the operation of matrix multiplication. Define
, where

is the Kronecker delta,

The equation just says
is defined as the matrix whose entry in
the row and

column is . In other words,
is the n · n matrix with ones down the main
diagonal

and zeros everywhere else. For example,

For all m · n matrices, the right identity is the matrix
, because for
all m · n

matrices A. Similarly, is the left identity, because
for all m · n
matrices. When

m ≠ n we cannot have a two -sided identity for matrix multiplication, because we
must

multiply on the left by an m · m matrix and on the right by an n · n matrix. So the
same

thing doesn't work on both sides. But for square n · n matrices, we do have a
two-sided

identity, just , because then
In an abuse of language, we usually call

the n · n identity matrix, even if we're using it on non- square matrices .

Now that we understand identities, let's discuss inverses, which have the
following prop-

erty: operating on something with its inverse should give you back the identity
of that

operation. For example, the** right inverse** of the number 3 under multiplication
is 1/3 , be-

cause , which is the multiplicative identity . Note that 1/3 is also its left inverse

because Since it works on both sides, it's called a two -sided inverse, or just
the

inverse of 3 under multiplication. Note that the inverse of 1 under
multiplication is just

itself, 1, because 1 ·1 = 1 = 1 ·1. Now consider addition. The (two-sided) inverse
of 3 is

-3 because -3 + 3 = 0 = 3 + (-3). Similarly, the inverse of the additive
identity 0 is just 0

itself because 0 + 0 = 0 = 0 + 0. Similarly, the inverse of
is
because

So the inverse of the identity is just the identity. This
is true in general: the inverse of the

identity is just the identity itself.

Inverses for addition and multiplication are easy enough, but what about
inverses for the

operation of function composition? This is a little more abstract. To figure out
what the

inverse is under composition, we first have to ask ourself what the identity is.
Consider all

functions with domain A and codomain B. The identity is just the function f : A
→ B

defined by f (x) = x, because it leaves what you put in unchanged. Now that we
know

what the identity is, what is the inverse of a function under function
composition? Like I

said in class, it is the function
such that
for all
x in B and

for all x in A. A simple example is
defined by f (x)
= x^{2}.

Then its inverse is defined by because and

for all x in (0,∞). That is, the functions g o f and f o g are both the

identity function, and therefore we can say that g = f^{ -1}.

Make sense? If you have any questions, please let me know.

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