Question

Let f(x) and g(x) be two linear functions. Note: a function is linear if there exists a,b∈R such that f(x)=ax+b. (a) Prove or disprove: f∘g=g∘f. (b) Prove or disprove: f∘g and g∘f are linear functions.

Answer 1

Let f(x) = ax + b and g(x) = cx + d.

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

= acx + ad + b

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

= acx + bc + d

In general, f(g(x)) ≠ g(f(x)).

(b) As ahown in (a), f(g(x)) and g(f(x)) are linear functions.