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

Question

2137 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-05T22:09:36

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

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.

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