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Currently, I am doing transformations on functions, with things like $y=f(x+2)$ representing a transformation of the original function 2 to the left, for example.

However, I am stuck on transforming inverse points. Given a point, say $(-3, 5)$ (made up) on $y=f(x)$, I want to transform it to the graph of $x+1=f(y-1)$.

I noticed the $x$ and $y$ have been switched, so I assume it's a inverse function, so my point is now at $(5, -3)$. I proceeded to isolate $x$, giving me $x=f(y-1)-1$. Then I decided I would need to translate the point 1 right, and 1 down, giving me $(6, -4)$.

However, the answer from the book (from the patterns I see, because I made up the point) is $(4, -2)$. From what I can tell, they left it at $x+1=f(y-1)$ and applied 1 left, and 1 up, the opposite of each to $x$ and $y$. However, I can't tell why.

Could anyone give a me a hint on what I have done wrong? Am I not allowed to move $-1$ over to the right side?

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I haven't looked at your geometric reasoning, but Algebraically, in order for $\color{Red}{x+1}=f(\color{Blue}{y-1})$ to correspond to the special values $\color{Red}5=f(\color{Blue}{-3})$, we need $\color{Red}{x+1}=\color{Red}5$ and $\color{Blue}{y-1}=\color{Blue}{-3}$, which when solved gives the textbook solution $x=4, y=-2$.


The part I don't follow in your argument is this:

Then I decided I would need to translate the point 1 right, and 1 down

I don't see why you would do that. Going from $x=\text{blah}$ to $x=\text{blah}-1$ involves translating 1 point to the left, not to the right. Similarly, replacing $y$ with $y-1$ in an equation will translate up, not down. So you should have gone $(-3,5)\mapsto (5,-3)\mapsto (4,-3)\mapsto (4,-2)$.

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Thank you. It seems like I made things a bit complicated. The reason why I performed everything backwards is that I thought the rules of a function in form $y=f(x-b)+a)$, where you apply a transformation $b$ units rights and $a$ units up (depending on the signs). Seems like instead it is just some algebra. Thanks again! (May it also be because when you take the inverse, things happen in the opposite order?) –  CuriousFr Feb 28 '12 at 3:04
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