question about inverse functions Functions $f$ and $g$ are defined by
$$ f: x\mapsto  2x+1$$
$$g: x \mapsto \dfrac{2x +1}{x+3}$$
(i)     Solve the equation $gf(x) = x $
(iii)   Show that the equation  $g^{-1} (x) = x$ has no solutions
I need help with these.

$$ f(x)=2x+1,g(x)= \frac{2x +1}{x+3}$$
$$g(f(x)) =\frac{2(2x+1) +1}{2x+1+3}= x$$
$$\frac{4x+3}{2x+4}= x,x\neq-2\Rightarrow2x^2+4x=4x+3,x^2=3/2,x=\pm\sqrt{3/2}$$
$$g(x)=y= \frac{2x +1}{x+3}$$
$$x= \frac{2y +1}{y+3},xy+3x=2y+1,y(x-2)=1-3x,y=\frac{1-3x}{x-2},x\neq2$$
$$g^{-1}(x)=\frac{1-3x}{x-2}=x\Rightarrow x^2+x-1=0,x=\frac{-1\pm\sqrt{5}}{2}$$
