For what values of x is the function f(x) = (4x+3)/(6x-4) invertible? Also, find the inverse…

$f(x)=\frac{4x+3}{6x-4}$ is defined when $6x-4\neq0$, i.e. $x\neq\frac{2}{3}$. It is one-one: $f(x_1)=f(x_2)\Rightarrow (4x_1+3)(6x_2-4)=(4x_2+3)(6x_1-4)\Rightarrow -16x_1+18x_2=-16x_2+18x_1\Rightarrow x_1=x_2$. Its range is $\mathbb{R}-\{\frac{2}{3}\}$ (the value $\frac{2}{3}$ is never reached), so $f:\mathbb{R}-\{\frac{2}{3}\}\to\mathbb{R}-\{\frac{2}{3}\}$ is invertible. Let $y=\frac{4x+3}{6x-4}$. Then $6xy-4y=4x+3$, so $x(6y-4)=4y+3$, $x=\frac{4y+3}{6y-4}$.