Apply g first: g(3) = 3² = 9. Then apply f to that result: f(9) = 2(9) + 1 = 19. A common error is applying f first: f(3) = 7, then g(7) = 49 — that computes g(f(3)), not f(g(3)). The order the functions are written in the notation tells you which to apply first (innermost first).
Question 2 True / False
For any two functions f and g, f(g(x)) generally equals g(f(x)).
TTrue
FFalse
Answer: False
Composition is not commutative in general. Using the same f(x) = 2x + 1 and g(x) = x²: f(g(x)) = 2x² + 1, but g(f(x)) = (2x+1)² = 4x² + 4x + 1 — clearly different expressions. There are special cases where they are equal (e.g., inverse functions), but this is not true in general.
Question 3 Short Answer
In the expression f(g(x)), which function is applied first, and how does the notation signal this?
Think about your answer, then reveal below.
Model answer: g is applied first. The notation signals this because g(x) is the input to f — it sits inside the parentheses of f. You always evaluate from the inside out.
Reading f(g(x)) literally: 'f applied to g(x).' To compute f of something, you first need to know what that something is — so you must evaluate g(x) before you can evaluate f. Inside-out evaluation is the key principle of function composition.