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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.

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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.

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