What is the range of the inverse of the function f(x) = (x + 2)/(x - 2)?

To determine the range of the inverse of the function f(x) = (x + 2)/(x - 2), we first need to examine the behavior of the original function. The function is defined for all real numbers except for x = 2, where it becomes undefined due to division by zero.

To find the range of f(x), we can express f(x) in terms of y and solve for x. Rearranging gives us:

y = (x + 2)/(x - 2)

Multiplying both sides by (x - 2) leads to:

y(x - 2) = x + 2

This can be rearranged to:

yx - 2y = x + 2

Now, grouping the x terms:

yx - x = 2y + 2

Factoring out x yields:

x(y - 1) = 2y + 2

From this, we can solve for x:

x = (2y + 2)/(y - 1)

The expression shows that x is undefined when y = 1, meaning that f(x) cannot take the value y = 1. Thus, the range of f(x) consists of all real numbers