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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) = x2. 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 identity 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) = x2.
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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