Question

If $y=f\left(x\right)=\frac{6x-5}{5x-6},$, prove that $f\left(y\right)=x,x\ne \frac{6}{5}.$

Answer 1

$y=f\left(x\right)=\frac{6x-5}{5x-6}$

$f\left(y\right)=f\left(f\right(x\left)\right)=\frac{6\left(\frac{6x-5}{5x-6}\right)-5}{5\left(\frac{6x-5}{5x-6}\right)-6}$

$=\frac{6(6x-5)-5(5x-6)}{5(6x-5)-6(5x-6)}$

$=\frac{36x-30-25x+30}{30x-25-30x+36}$

$=\frac{9x}{9}$

$f\left(y\right)=x$