Question

We call a function f : R→R addition-friendly if for all a, b ∈R, f (a + b) = f (a) + f (b).

(a) Determine whether or not each of the following functions are addition-friendly. Prove
your answers.
(i) f (x) = 2x
(ii) g(x) = 2x + 3
(b) If f and g are both addition-friendly, is f ◦g addition-friendly? Prove your answer or
give a counterexample.

Answer 1

Upon examination, f(x) = 2x is addition-friendly because it satisfies the addition-friendly condition, while g(x) = 2x + 3 does not. As g(x) is not addition-friendly, we cannot confirm that the composition f ○ g is addition-friendly without further investigation.

Step-by-step explanation:

To determine whether the functions f(x) = 2x and g(x) = 2x + 3 are addition-friendly, we have to check if they satisfy the condition f(a + b) = f(a) + f(b).

• For f(x) = 2x: f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b). This confirms that f(x) is addition-friendly.
• For g(x) = 2x + 3: g(a + b) = 2(a + b) + 3 = 2a + 2b + 3 ≠ (2a + 3) + (2b + 3) = g(a) + g(b). Therefore, g(x) is not addition-friendly.

For the composition f ○ g, if both f and g were addition-friendly, then (f ○ g)(a + b) should equal (f ○ g)(a) + (f ○ g)(b). However, g(x) is not addition-friendly, so we cannot assert that f ○ g will be addition-friendly without further examination. In this case, a counterexample or additional proof is required, but because g(x) is not addition-friendly, f ○ g is not necessarily addition-friendly.