Page 166 - Discrete Mathematics and Its Applications

2.3 Functions 145

Suppose that f : A → B.
To show that f is injective Show that if f(x) = f(y) for arbitrary x, y ∈ A with x = y,
then x = y.
To show that f is not injective Find particular elements x, y ∈ A such that x = y and
f(x) = f(y).
To show that f is surjective Consider an arbitrary element y ∈ B and find an element x ∈ A
such that f(x) = y.
To show that f is not surjective Find a particular y ∈ B such that f(x) = y for all x ∈ A.

Inverse Functions and Compositions of Functions

Now consider a one-to-one correspondence f from the set A to the set B. Because f is an onto
function, every element of B is the image of some element in A. Furthermore, because f is also
a one-to-one function, every element of B is the image of a unique element of A. Consequently,
we can define a new function from B to A that reverses the correspondence given by f . This
leads to Definition 9.

DEFINITION 9 Let f be a one-to-one correspondence from the set A to the set B. The inverse function of
f is the function that assigns to an element b belonging to B the unique element a in A
such that f(a) = b. The inverse function of f is denoted by f −1 . Hence, f −1 (b) = a when
f(a) = b.

Remark: Be sure not to confuse the function f −1 with the function 1/f , which is the function
that assigns to each x in the domain the value 1/f (x). Notice that the latter makes sense only
when f(x) is a non-zero real number.

Figure 6 illustrates the concept of an inverse function.
If a function f is not a one-to-one correspondence, we cannot define an inverse function of
f . When f is not a one-to-one correspondence, either it is not one-to-one or it is not onto. If
f is not one-to-one, some element b in the codomain is the image of more than one element in
the domain. If f is not onto, for some element b in the codomain, no element a in the domain
exists for which f(a) = b. Consequently, if f is not a one-to-one correspondence, we cannot
assign to each element b in the codomain a unique element a in the domain such that f(a) = b
(because for some b there is either more than one such a or no such a).
A one-to-one correspondence is called invertible because we can define an inverse of this
function. A function is not invertible if it is not a one-to-one correspondence, because the
inverse of such a function does not exist.

–1
f (b)

–1
a = f (b) f(a) b = f(a)

f –1
A B
f

FIGURE 6 The Function f −1 Is the Inverse of Function f .