what is the inverse of g(x)=2/(3−x)+2, I am getting a bit bamboozled :(

Question

thomas5267 · Answer

$$
\begin{align*}
y&=\frac{2}{3-x}+2\
y-2&=\frac{2}{3-x}\
(y-2)(3-x)&=2\
3y-xy-6+2x&=2\
-xy+2x&=8-3y
\end{align*}
$$
Factor out x on the left hand side and solve for x.

anonymous · Answer

so $x=(8-3y)/(2-y)$

e.mccormick · Answer

Yes. Visually, an inverse is the reflection across the line $y=x$. That goes along with what I said in the last question about how $f(x)$ and $f^{-1}(x)$ swap the domain and range. https://www.desmos.com/calculator/5ktn1vxrdq Only thing I would have done differently than thomas5267 is swap the x and y at the start. Then solve for y. It is a minor conceptual difference that can help some people, but does not really matter in the end.

anonymous · Answer

so my asymptotes swap as does the range and domain?

e.mccormick · Answer

Yah, just like the other function.  When doing an inverse remember to reverse:  The stuff swaps.  While you may still need to show the math to get the work correct, it makes for an easy sanity check.

anonymous · Answer

I just have so much trouble getting the inverse of the function, it does my head in :(