Basic Definitions
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(c) For x E IR, let Sqrt(x) be the non-negative square root of x. Then, Sqrt is a partial
function, since Sqrt(x) is undefined for x < 0. The domain of definition of Sqrt is iR,
and its codomain is R. The range of Sqrt is [0, oo).
Let G be a subset of R. G is the graph of a partial function if, whenever x0 E X,
the vertical line x = xo intersects G in at most one point. We call this the vertical line
test for a partial function. Figure 4.8 shows a subset of R x R that is not a function,
because the vertical line x = -1 does not cross the graph. Sqrt is a partial function,
since no vertical line defined by an element of its domain crosses the graph more than
once.
y
2.5
2
1.5
0.5
-2
2
4
6!
8
Figure 4.8 Graph of partial function Sqrt.
Whether a partial function is a total function depends on what the domain of definition
is defined to be. For example, it was noted that Sqrt is a partial function from JR to IR. If we
declare the domain of definition to be just the set [0, co), then Sqrt is a total function.
4.1.8 1-1 and Onto Functions
Several special types of functions have turned out to be especially important. For example, the intuitive notion of counting will be formalized using the properties of functions
introduced in this section.
Definition 6. Let F : X -- Y be a function. F is 1-1 if, for each y E Y, there is, at most,
one x E X such that F(x) = y.
Example 16.
(a) Let F :R -- R be a function defined as F(x) = 2x. F is 1-1.
(b) Let G N -- N be a function defined as G(n) = 2n
2 + 1. G is not 1-1.
Solution.
(a) Since F(xl) = F(x 2 ) means 2xl = 2x2, it follows that xl = X2 and F is 1-1.
(b) Since G(2) = G(-2), the function G is not 1-1.
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