<s>[INST] Points of Intersection between a function and its inverse Say we have this function: $g(x) = 1+ \frac{1}{x+1}$ for $x>-1$. 
The question asks to solve for $g(x) = g^{-1}(x)$. I intuitively thought to first calculate the inverse function's equation which turned out to be $y=\frac{-x+2}{x-1}$ and then to simply equate it with the original function (which I thought was the normal way to find POI's of functions). However, while this did produce an answer ($x=0,-2$) they were not correct (the answer is $x=\sqrt{2}$.Could I have some direction on this please.
 [/INST] Your method is correct, but you made a mistake when finding the inverse of $g$. To see why, note that $g(0)=2$ but
$$
-\frac{(2)+2}{(2)-1} = -4 \ne 0.
$$
It should be
$$
g^{-1}(x) = \frac{2-x}{x-1}.
$$
Let's verify that this is correct. Note that
$$
(g\circ g^{-1})(x) = 1+\frac{1}{\frac{2-x}{x-1}+1}
= 1+\frac{1}{\frac{1}{x-1}} = 1+(x-1)=x
$$
and
$$
(g^{-1}\circ g(x) = \frac{2-(1+\frac{1}{x+1})}{(1+\frac{1}{x+1})-1}
= \frac{1-\frac{1}{x+1}}{\frac{1}{x+1}}
= \frac{1}{\frac{1}{x+1}} -1=(x+1)-1=x.
$$

Continuing with the solution, we need to solve $g(x)=g^{-1}(x)$. That is:
\begin{align*}
1+\frac{1}{x+1} &= \frac{2-x}{x-1} \\
(x+1)(x-1) + (x-1) &= (x+1)(2-x) \\
(x-1)(x+2) &= (x+1)(2-x) \\
x^2+x-2 &= -x^2+x+2 \\
2x^2-4&=0 \\
x^2 &= 2 \\
x &= \pm\sqrt{2}
\end{align*}
Since the domain excludes $x=-\sqrt{2}$, we are left with only $\sqrt{2}$.
</s>