Try our Free Online Math Solver!

Functions and Inverse Functions
Asterisks denote definitions which are not in the
textbook.
1. We review the definitions leading up to the textbook's definition of inverse
function.
(a) *A relation is a set of ordered pairs .
(b) If there are sets R, A, and B such that
R ⊆ A × B,
then R is said to be a relation between A and B.
(Clearly, if R is a relation between A and B, then R is a relation as defined in
(a).)
(c) *The domain of a relation R, denoted by dom(R), is defined by
(d) *A relation R is called a function, or is said to have the function
property , iff
∀ a, b, c, and d, if (a, b) ∈ R and (c, d) ∈ R and a = c, then b = d.
In other words, a relation R is a function iff
if (a, b) ∈ R and (a, d) ∈ R then b = d.
(e) Given sets A and B, a relation F is said to be a function from A to B
iff
F is a relation between A and B;
F has the function property ;
dom(F ) = A (note that the fact that F is a relation between A and B implies
that ran(F) ⊆ B ).
The notation F : A → B means that F is a function from A to B, although the
exact verbal
interpretation of the symbols F : A → B may differ slightly depending on the
context.
(f) Theorem. *For a function F : A → B, define the relation F^{1} by
(I) (b, a) ∈ F^{1} iff (a, b) ∈ F , (I)
(i.e.,
z ∈ F^{1} iff ∃ a, b z = (b, a) and (a, b) ∈
F )
If the function F : A → B is bijective, then F ^{1} is a
function from B to A. (You should be
able to prove this.)
(g) The definition of inverse function given in the textbook can be expressed in
the following way:
If a function F : A → B is bijective, then the function F^{1}: B → A
defined above is called the
inverse function of F : A → B.
*To distinguish this concept from another inverse function concept discussed
below, we will
temporarily refer to this as the Linverse function of F : A → B.
(The letter “L” refers to the author of the textbook.)
*I.e., a function G : B → A is the Linverse function of a bijective function F
: A → B iff
dom(G) = B and G is the inverse relation F^{1} of the relation F as defined in
(I) above.
(h) *Expressed in standard function notation instead of with the use of ordered
pairs , equation (I)
above gives the following defining condition for the Linverse:
For every a in A and b in B ,
(I') F^{1}(b) = a iff F(a) = b.
We will give another definition of “inverse function”, and
then we will comment on the definitions and
the relationships between them.
2. *Given a function f : A → B, a function g : B → A is called an inverse
of the function f : A → B
iff
∀ x in A, g (f (x)) = x,
and
∀ y in B, f (g (y)) = y.
We will temporarily refer to such a function g : B → A as an
algebraic inverse function of
f : A → B .
It should be clear that this definition is equivalent to the following :
For any set S, let id_{S} denote the identity function on S.
Given a function f : A → B, a function g : B → A is called an
( algebraic )inverse of the
function f : A → B iff
g o f = id_{A}
and
f o g = id_{B} .
Comments:
Uniqueness. It is clear that the Linverse of a function is unique, if it
exists, since the Linverse
function is obtained in a particular way from the original function. On the
other hand, if the inverse
function is defined as an algebraic inverse, then it has to be proved that if
such an inverse function
exists, then it is unique.
Equivalence of L inverse and algebraic inverse. You should be able to
prove (this is related to
Ex. 7.29 in the textbook) that the Linverse idea and the algebraic inverse idea
are equivalent in the
following sense:
A function f : A → B has an algebraic inverse function iff f : A → B is
bijective.
A function g : B → A is the Linverse function of f : A → B iff g : B → A is an
algebraic inverse
function of f : A → B.
Uniqueness. Note that this implies that if an algebraicinverse function
exists, then it is unique.
Prev  Next 