Question

14 mark for review the function f is defined by f(x) = √(4 - x²) for -2 ≤ x ≤ 0. which of the following expressions defines f⁻¹(x)? a -√(4 - x²) for -2 ≤ x ≤ 0 b √(4 - x²) for -2 ≤ x ≤ 0 c -√(4 - x²) for 0 ≤ x ≤ 2 d √(4 - x²) for 0 ≤ x ≤ 2

Explanation:

Step1: Set y = f(x)

$y = \sqrt{4 - x^2}$

Step2: Swap x and y for inverse

$x = \sqrt{4 - y^2}$

Step3: Square both sides

$x^2 = 4 - y^2$

Step4: Solve for y²

$y^2 = 4 - x^2$

Step5: Determine y sign (original x ≤0)

Original domain: $-2 ≤ x ≤0$, so inverse range is $-2 ≤ y ≤0$. Thus $y = -\sqrt{4 - x^2}$

Step6: Find inverse domain

Original range: $0 ≤ y ≤2$, so inverse domain is $0 ≤ x ≤2$

Answer:

C. $-\sqrt{4 - x^2}$ for $0 ≤ x ≤ 2$