Condition : \(x\ne-5\)Since the function f is the inverse of g(x), \(f\left(g\left(x\right)\right)=x\) or \(f\left(\dfrac{2x-3}{x+5}\right)=x\) . Denote \(f\left(4\right)=y\), then :

\(\dfrac{2y-3}{y+5}=4\Leftrightarrow2y-3=4\left(y+5\right)\Leftrightarrow2y-3=4y+20\)

\(\Leftrightarrow-3-20=4y-2y\Leftrightarrow2y=-23\Leftrightarrow y=-\dfrac{23}{2}\)

So, \(f\left(4\right)=-\dfrac{23}{2}\)