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Answer :
To determine if two functions are inverses, compose them in both orders and simplify. If upon simplification, you get the identity function, then they are inverses of each other.
To determine whether two functions are inverses of each other, you can compose them and see if the result is the identity function, which means that for function f and its inverse g, f(g(x)) = x and g(f(x)) = x should hold true. If we denote the first function as f(x) = 2/(x-3) and the second function as g(x) = (2+3x)/x, we would compute f(g(x)) and g(f(x)) as follows:
- Calculate f(g(x)): f((2+3x)/x) = 2 / (((2+3x)/x) - 3)
- Calculate g(f(x)): g(2/(x-3)) = (2+3(2/(x-3)))/(2/(x-3))
After simplifying both expressions, if we end up with f(g(x)) = x and g(f(x)) = x, the functions are inverses of each other. Otherwise, they are not. You would check for any simplifications that result in x, indicating that they are indeed inverses.
This method utilizes the properties of inverses and function composition, which are key concepts in understanding advanced algebra and functions.
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