Question

How do you decide whether the relation `f(x)= x/ [(x+2)(x-2)]` defines a function?

Answer 1

This formula does define a function since it defines a unique value for any `x` in the (implicit) domain.

Explanation:

For any Real value of `x` except `x = +-2`, the formula:

`f(x) = x/((x+2)(x-2))`

uniquely defines a unique Real value.

When `x = +-2`, the denominator of `f(x)` is zero, so `f(x)` is undefined and these values of `x` are not in the domain.

So the given formula defines a function on the (implicit) domain:

`(-oo, -2) uu (-2, 2) uu (2, oo)`

The graph of `f(x)` satisfies the vertical line test.

graph{x/((x+2)(x-2)) [-10, 10, -5, 5]}

Any vertical line only intersects `f(x)` at one point at most.