Functions: Question 6

Worked solution

Step 1: Identify the condition for the function to be defined

A square root is only defined (as a real number) when the expression underneath it is non-negative. So we need: $3x - 12 \ge 0$

Step 2: Solve the inequality

Add $12$ to both sides: $3x \ge 12$

Divide both sides by $3$ (a positive number, so the inequality direction is unchanged): $x \ge 4$

Step 3: Check the boundary value

At $x = 4$: $f(4) = \sqrt{3(4)-12} = \sqrt{0} = 0$, which is defined. So $x=4$ is included in the domain. The inequality should not be strict.

Why the other options are wrong

• B ($x\le4$): reverses the inequality direction.
• C ($x\ge-4$): forgets to divide by the coefficient $3$.
• D ($x>4$): wrongly excludes the boundary value $x=4$, where $f(x)=\sqrt{0}=0$ is perfectly well defined.

Final answer

$\boxed{x \ge 4} \quad \text{(Option A)}$