Questions: Suppose that f(x) = sqrt(1-x^2) for 0 ≤ x ≤ 1. Find f^(-1)(x).

Solution Steps

We start with the function defined as \( f(x) = \sqrt{1 - x^{2}} \) for \( 0 \leq x \leq 1 \).

To find the inverse function \( f^{-1}(x) \), we solve the equation \( y = \sqrt{1 - x^{2}} \) for \( x \).

Squaring both sides gives us: \[ y^{2} = 1 - x^{2} \] Rearranging this equation leads to: \[ x^{2} = 1 - y^{2} \] Taking the square root of both sides, we find: \[ x = \sqrt{1 - y^{2}} \quad \text{or} \quad x = -\sqrt{1 - y^{2}} \]

Since the original function \( f(x) \) is defined for \( 0 \leq x \leq 1 \), the valid inverse function is: \[ f^{-1}(x) = \sqrt{1 - x^{2}} \] However, we also have the negative root, which gives us: \[ f^{-1}(x) = -\sqrt{1 - x^{2}} \] Thus, the complete inverse function is: \[ f^{-1}(x) = -\sqrt{1 - x^{2}}, \sqrt{1 - x^{2}} \]

Final Answer