A function f is defined by $f(x) = \displaystyle\frac{2x-3}{x-1}, x≠1$. Solve the equation |$f^{-1}(x)$| = 1 + $f$-1$(x)$.
I first found out the inverse and equated for the left hand side the negative of the inverse and then solved. However, I got the wrong answer and was unsure why.
Finding the inverse is actually unnecessary. The only solution to the equation to
$$|z| = 1 + z$$
is $z = -\frac{1}{2}$ (this is because the only way absolute value gives you a different number from the input is when the input is negative).
In other words we have that
$$f^{-1}(x) = -\frac{1}{2} \implies x = f\left(-\frac{1}{2}\right) = \frac{8}{3}$$