# Calculating inverse - what about assumptions?

Can someone verify this and answer my questions? I've chosen simple function on purpose. I've also added my paper from an exam at the very end.

Find inverse function $$f^{-1}$$ to function $$f(x)= \frac{x+1}{x-3}$$. Show the domain of both $$f$$ and $$f^{-1}$$.

Domain of $$f$$:

$$x-3 \neq 0$$

$$x \neq 3$$

$$D_{f} = \mathbb{R}\backslash\{3\}$$

Finding inverse:

$$y = \frac{x+1}{x-3}$$

$$x \longleftrightarrow y$$

$$x = \frac{y+1}{y-3} \quad \quad /*(y-3)\quad \quad$$ QUESTION 1

$$xy-3x=y+1$$

$$xy-y=3x+1$$

$$y(x-1)=3x+1 \quad \quad /:(x-1) \quad \quad$$ QUESTION 2

$$y=\frac{3x+1}{x-1}$$

$$f^{-1}(x) = \frac{3x+1}{x-1}$$

Domain of inverse:

$$x = \mathbb{R}\backslash\{1\}$$

I tagged "question 1" and "question 2" above.

Question 1: when multiplying both sides by $$(y-3)$$, should I make an assumption that $$y-3 > 0 \Rightarrow y > 3$$? Because what if y was negative or $$0?$$ Then the sign would change... Why should I make or why should I not make such assumption?

Question 2: when dividing both sides by $$(x-1)$$, should I make an assumption that $$x - 1 > 0 \Rightarrow x > 1$$? Because what if I'm dividing both sides by $$(x-1)$$ and $$(x-1)$$ was negative or $$0?$$ Then the sign would change... Why should I make or why should I not make such assumption?

Question 3: how is it possible that the assumptions from Question 1 and/or Question 2 don't affect the final domain of $$f^{-1}(x)$$?

Side note: For example here (link to my paper: https://i.imgur.com/zkJxAOD.jpg ) I was told to make an assumptions. I am really confused now when should I make an assumptions and when not to...

Thanks for any help.

Q1: No, because this is an equation, not an inequality. More details: $$y\neq 3$$ is assumed since $$(y-3)$$ is in the denominator. So we can freely multiply by $$(y-3).$$
Q2: Similarly, we are no signs to care about. Though when dividing by $$(x-1)$$ we should make assumption $$x\neq 1.$$
Q3: The final domain is affected, because $$1$$ is excluded.