Question

For any linear function f(x) = mx + b, when does kf(x) = f(kx)?

Answer 1

Only when b = 0 or k = 1.

If f(x) = mx + b, then

f (kx) = m (kx) + b = mkx + b

while

k f(x) = k (mx + b) = kmx + kb

The two expression are identical only if

kb = b   ===>   kb - b = 0   ===>   b (k - 1) = 0   ===>   b = 0 or k = 1