Demonstration of finding inverse of f(x)=Sqrt[x+1]  algebraically, and plotting the function and its inverse simultaneously.

f[x_]:=Sqrt[x+1]
f[x]
Plot[f[x],{x,-1,9},PlotRange->{-1,9}]

g[y_]:=-1+y^2
g[x]

Plot[{f[x], g[x]}, {x, -1, 9},PlotRange->{-1,9} ]

Plot[Sqrt[1 + x], {x, -1, 9}, PlotStyle -> Blue, {}]
plotOrig=%;

Plot[-1 + x^2, {x, 0, 9}, PlotStyle -> Red, {}]
plotInverse=%;

Plot[x,{x,-1,9},PlotStyle->Dashing[Medium]]
plotIdentity=%;

Show[{plotOrig,plotInverse,plotIdentity},PlotRange->{-1,9},AspectRatio->1]