Let f(x) = ax + b and g(x) = cx + d . The condition f(g(x)) = g(f(x)) holds for all x if and only if ...

Step-by-step solution

Given f(x) = ax + b and g(x) = cx + d . Substituting g(x) into f(x) : f(g(x)) = a(cx + d) + b = acx + ad + b Substituting f(x) into g(x) : g(f(x)) = c(ax + b) + d = acx + cb + d For f(g(x)) = g(f(x)) to hold for all x , the corresponding coefficients and constant terms must be equal: acx + ad + b = acx + cb + d ad + b = cb + d Now, evaluating f(d) and g(b) : f(d) = ad + b g(b) = cb + d Therefore, the required condition is f(d) = g(b) . Answer: f(d) = g(b)