If f (x) = ax+b and g(x) = cx + d, then f (g(x)) = g( f (x)) if

$f (g(x)) = ag(x)+ b = a(cx + d)+ b = acx + ad +b$

$g( f (x)) = cf (x)+ d = c(ax +b)+ d = acx +bc + d$

$∴ f (g(x)) = g( f (x))$ if $ad + b = bc + d$

i.e. if $f (d) = g(b)$