Page 169 - A Course in Linear Algebra with Applications

6.1: Functions Defined on Sets 153

of linear algebra is F : M TO)n(R) —> R defined by F(A) =
det(A), that is, the determinant function.
A very simple, but nonetheless important, example of a
function is the identity function on a set X; this is the function

l x : X -> X
which leaves every element of the set X fixed, that is, lx(x) =
x for all elements x of X.
Next, three important special types of function will be
introduced. A function F : X —> Y is said to be injective (or
one-one) if distinct elements of X always have distinct images
under F, that is, if the equation F(xi) = F{x2) implies that
X\ = X2- On the other hand, F is said to be surjective (or
onto) if every element y of Y is the image under F of at least
one element of X, that is, if y = F(x) for some x in X Finally,
F is said to be bijective (or a one-one correspondence) if it is
both injective and surjective.
We need to give some examples to illustrate these con-
cepts. For convenience these will be real-valued functions of
a real variable x.

Example 6.1.1
X
Define F x : R -• R by the rule F ±(x) = 2 . Then F x is
injective since 2 X = 2 y clearly implies that x — y. But i*\
cannot be surjective since 2 X is always positive and so, for
example, 0 is not the image of any element under F.

Example 6.1.2
2
Define a function F2 : R —> R by F 2(x) = x (x — 1). Here
F2 is not injective; indeed ^ ( 0 ) = 0 = ^ ( 1 ) . However F2 is
2
surjective since the expression x (x — 1) assumes all real values
as x varies. The best way to see this is to draw the graph of
2
the function y = x (x — 1) and observe that it extends over
the entire y-axis.