Algebra Know-It-ALL

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.