Background: I was trying to solving this question:
Find $\frac{1}{x}$, if $x \in(-1,3)-\{0\}$.
I know that I can solve the question by drawing the graph of the function, $\frac{1}{x}$, then I should find the values of the function at $-1$ and $3$, then I should "look" for the values the function assumes between the two values. But this method is impractical; what if the function is not easily "graphable"? How should one proceed then? That's why I tried to find algebraic method to solve this question and hence this type of questions.
I tried to solve it, algebraically, like this:
I know that domain of a inverse function is equal to the range of the function. So to find the range of $f(x)\frac{1}{x}$, with the restriction, I should first find the inverse of the function.
The inverse of the function is $f^{-1}(x)=\frac{1}{x}$, with $x, f^{-1}(x)\not= 0 $ and $f^{-1}(x) \in(-1,3)$.
Now all I have to do is to find Domain of this new function. But I don't know how to proceed from here. And this method is also impractical, because not every function is one to one function.
Question: The question is how to find range of a function with restricted domain, algebraically? Note that I am not asking how to solve that particular question of finding $\frac{1}{x}…$, but what I am asking is methods to solve this type of questions, algebraically.