Page 173 - A Course in Linear Algebra with Applications

6.1: Functions Defined on Sets 157

Theorem 6.1.4
Every bijective function F : X —> Y has a unique inverse
function.

Proof
Suppose that F has two inverse functions, say G\ and G 2.
Then {G x o F) o G 2 = lx ° G 2 = G 2 by 6.1.2. On the other
hand, by 6.1.1 this function is also equal to G\ o (F o G 2) =
G i o l y = Gi. Thus Gi = G 2 .

Because of this result it is unambiguous to denote the
inverse of a bijective function F : X —•> Y by

F~ l : Y -> X.

To conclude this brief account of the elementary theory of
functions, we record two frequently used results about inverse
functions.
Theorem 6.1.5
l
(a) If F : X —> Y is an invertible function, then F~ is
invertible with inverse F.
(b) IfF:X^YandG:U^X are invertible
functions, then the function F o G : U —> Y
1
is invertible and its inverse is G~ x o F" .
Proof
l x
Since F o F~ = 1 Y and F~ o F = l x, it follows that F is
x
the inverse of F~ . For the second statement it is enough to
check that when G - 1 o F _ 1 is composed with F o G on both
sides, identity functions result. To prove this simply apply the
associative law twice.