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

To determine the domain of the inverse of the function ( f(x) = \frac{x + 2}{x - 2} ), we first need to understand the function itself and its range. The function is a rational function, and like all rational functions, it has potential restrictions in its domain.

The domain of ( f(x) ) is all real numbers except where the denominator equals zero. For this function:

[

x - 2 = 0

]

This gives ( x = 2 ). Therefore, the domain of ( f(x) ) is all real numbers except ( x = 2 ).

Next, we need to find the range of ( f(x) ) in order to determine the domain of the inverse. As ( x ) approaches 2 from either side, ( f(x) ) approaches infinity, and as ( x ) goes to positive or negative infinity, the function approaches 1. By analyzing the function, we can conclude that it can take all real values except for 1 because there is a horizontal asymptote at ( y = 1 ).

Given that the range of ( f(x) ) excludes ( y = 1 \