PLEASE PLEASE HELP Let fx ax b and gx cx d where a b c a

PLEASE PLEASE HELP

Let f(x) = ax + b and g(x) = cx + d where a, b, c, and d are constants. Determine the necessary and sufficient conditions for f middot g = g middot f.

Solution

Given y = f(x)
y = g(x)

f(x)og(x) = f(g(x))

g(x)of(x) = g(f(x))

f(x) = ax + b
g(x) = cx + d

fog = a(cx + d) + b = acx + ad + b

gof = c(ax + b) + d = acx + bc + d

Equating them:

acx + ad + b = acx + bc + d

acx cancel on both sides:

ad + b = bc + d

ad + b = bc + d

is the necessary condition for : fog = fog

A quick note:

fog = f(g(x)), you put g(x) everywhere you see an x in f.

gof = g(f(x)), you put f(x) everywhere you see an x in g.