A familiar functional equation f ( a x + b ) = c f ( x ) will be solved in the class of functions f : ℝ → ℝ . Applying this result we will investigate the Hyers-Ulam-Rassias stability problem of the generalized additive Cauchy equation f ( a 1 x 1 + ⋯ + a m x m + x 0 ) = ∑ i = 1 m b i f ( a i 1 x 1 + ⋯ + a i m x m ) in connection with the question of Rassias and Tabor.