Functions: Question 9

Worked solution

Step 1: Understand the order of composition

$gf(-2)$ means $g(f(-2))$, apply $f$ first, then apply $g$ to that result.

Step 2: Find $f(-2)$

$f(-2) = (-2)^2 - 1 = 4 - 1 = 3$

Step 3: Apply $g$ to this result

$g(3) = 3 - 3 = 0$

So $gf(-2) = 0$.

Why the other options are wrong

• B ($3$): this is just $f(-2)$, the working stops one step too early, forgetting to apply $g$.
• C ($24$): this comes from computing $fg(-2)$ instead, i.e. applying the functions in the wrong order: $g(-2)=5$, then $f(5)=25-1=24$.
• D ($8$): this comes from a sign error, treating $(-2)^2$ as $-4$ so that $f(-2)$ is wrongly found to be $-5$, then $g(-5)=3-(-5)=8$.

Final answer

$\boxed{gf(-2) = 0} \quad \text{(Option A)}$