how to prove two functions are inverses
plug inverse function into original function and see if it solves for x
To prove that two functions are inverses, we should demonstrate that they undo each other, meaning that when we apply one function to the input and then apply the second function to the resulting output, we obtain the original input. Symbolically, for two functions f and g, we need to prove that:
f(g(x)) = x AND g(f(x)) = x
Here are the steps to follow in order to prove that two functions f and g are inverses:
1. Start by verifying that f(g(x))= x. This means you have to substitute in the definition of g(x) for all x values in f(x), simplify and simplify again to show that it equals x.
2. Similarly, verify that g(f(x)) = x. Now, you have to substitute in the definition of f(x) for all x values in g(x), simplify, and then simplify again to demonstrate that it equals x.
3. If f(g(x))= x AND g(f(x))=x, you have proven that f and g are inverses of each other.
If you need to simplify and can’t see an obvious end result, consider working backwards, substituting in certain values into the function that you know have a particular result. If you can reverse the operation to get back there, then it gives you a sense of how to simplify correctly.
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