Algebra Know-It-ALL

This is another example of a function. If we call it g, we can write

g(v)=v^2

For every value of v in the domain of g, there is exactly one value of w, which we can also call
g(v), in the range. But the reverse is not true. For every nonzero value of w in the range of g,
there are two values of v in the domain. These two values are additive inverses (negatives) of
each other. For example, if w= 49, then v= 7 or v=−7. This is not a problem; a relation can
be many-to-one and still be a function. The trouble happens when a relation is one-to-many.
Then it can’t be a function.
Our function g is not injective because it’s two-to-one, not one-to-one (except when
v= 0). Therefore, it cannot be bijective. The function g is surjective, however, because every
possible value in its range (the set of nonnegative reals) is accounted for. In formal language
we say, “For any nonnegative real number w in the range of g, there is at least one v in the
domain such that g(v)=w.”

Cube the input

Here’s another relation. Let’s call the independent variable t and the dependent variable u. The
domain and range are both the entire set of reals. The equation is

u=t^3

This is a function. If we call it h, then

h(t)=t^3

For every value of t in the domain of h, there is exactly one value of u in the range. The reverse
is also true. For every value of u in the range of h, there is exactly one t in the domain.
This function is one-to-one, so it’s injective. It maps onto the entire range, so it’s surjective
as well. That means h is a bijection.

Are you confused?

As with relations, graphing can be useful when you want to see how functions map values of an independ-
ent variable into values of a dependent variable. Graphs of the above three functions are shown in the next
chapter.

Here’s a challenge!

With any relation, you can transpose the values of the independent and dependent variables while leav-
ing their names the same. You can also transpose the domain and the range. When you do these things,
you get another relation, which is called the inverse relation (or simply the inverse if the context is clear).
The inverse of a relation is denoted by writing a superscript −1 after the name of the relation. If you have
h(z), its inverse is written h−^1 (z). The inverse of a function, however, is not necessarily another function!
Look again at the three functions f, g, and h above. The inverse of one of these functions is not a function.
Which one?

Examples of Functions 219