216 Mappings, Relations, and Functions
When we plug specific values of the independent variable x into this equation, we get results
such as:
- If x= 1/9, then (x,y)= (1/9,1/3) or (1/9,−1/3)
- If x= 1/4, then (x,y)= (1/4,1/2) or (1/4,−1/2)
- If x= 1, then (x,y)= (1,1) or (1,−1)
- If x= 4, then (x,y)= (4,2) or (4,−2)
- If x= 9, then (x,y)= (9,3) or (9,−3)
- If x= 0, then (x,y)= (0,0)
This mapping is clearly not an injection! For every nonzero value of x, there are two values of y.
But the mapping is onto the entire co-domain. No matter what real number y we choose, we
can square it and get a nonnegative real number x. That means the mapping is a surjection, so
we can call this relation a surjective relation.
A bijective relation
Let’s modify the relation in the preceding section by restricting the co-domain and range Y to
the set of nonnegative reals. Then we get this equation to represent it:
y=x1/2
When there is no sign in front of an expression raised to the 1/2 power, then by convention,
the 1/2 power indicates the nonnegative square root. Now there’s only one output value y for
every input value x. We’ve simply declared that all negative output values are invalid! Here are
some of the ordered pairs in this relation:
- If x= 1/9, then (x,y)= (1/9,1/3)
- If x= 1/4, then (x,y)= (1/4,1/2)
- If x= 1, then (x,y)= (1,1)
- If x= 4, then (x,y)= (4,2)
- If x= 9, then (x,y)= (9,3)
- If x= 0, then (x,y)= (0,0)
We now have a one-to-one mapping, and it’s also onto the entire co-domain. That means it’s
both injective and surjective. The equation
y=x1/2
represents a bijective relation within the set of nonnegative reals. No matter what nonnega-
tive real number x we plug into this relation, we get a unique nonnegative real number y
out of it. It also works the opposite way: No matter what nonnegative real y we want to
get out of this relation, we can find a unique nonnegative real x to plug in that will give
it to us.