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

1 Answer
May 5, 2016

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.