<s>[INST] If $y=f(x)=\frac{3x-5}{2x-m}$ find $m$ so that $f(y)=x$. Question: If $y=f(x)=\dfrac{3x-5}{2x-m}$ find $m$ so that $f(y)=x$.
We have $y=\dfrac{3\left(\frac{3x-5}{2x-m}\right)-5}{2\left(\frac{3x-5}{2x-m}\right)-m} $
How can I find $m$? It is given than $m=3$.
 [/INST] 1-method. The hyperbola has assymptotes: $y=\frac32$ and $x=\frac m2$. The inverse function must be symmetric with respect to the line $y=x$. For the hyperbola to be self inverse, the assymptotes must intersect at the line $y=x$, consequently, $m=3$.
2-method. Find the inverse function and equate to the original:
$$y=\frac{3x-5}{2x-m}\Rightarrow x=\frac{mx-5}{2x-3}\Rightarrow \\
y^{-1}=\frac{mx-5}{2x-3}=\frac{3x-5}{2x-m}\Rightarrow m=3.$$
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