TITLE: basic proof structuring question
QUESTION [1 upvotes]: If a question where to provide you with two functions and then ask you to "Find all values for which $f(x)=g(x)$",  how should the proof be structured?
What I am tempted to do is write $f(x)\stackrel{?}{=}g(x)$ and then to continue doing this until I get some results that show some restrictions.  Is there anything wrong with this tactic?  or is there a more standard/preferred method to go about proving such a problem?

REPLY [2 votes]: The problem can be translated as to find the necessary and sufficient conditions for $x$ to be such that $f(x)=g(x)$. It seems easier to begin with the necessary condition by investigating what properties $x$ must have if $f(x) = g(x)$. With this result, if further that all such $x$ satisfy $f(x) = g(x)$, then you are done.
For example, consider the problem of finding all $x$ such that $x+3 = 5$. To solve it, note that if $x+3 = 5$ then $x=2$. And if $x=2$ then $x+3 = 2+3 = 5$. So $\{ 2 \} = \{x \mid x+3=5 \}$. This exemplifies the above suggestion.
{"set_name": "stack_exchange", "score": 1, "question_id": 2325042}