Questions: f(x)=10-x^2, x ≥ 0 f^(-1)(x)=

Solution Steps

To find the inverse function \( f^{-1}(x) \), we need to solve the equation \( y = 10 - x^2 \) for \( x \) in terms of \( y \). This involves rearranging the equation to express \( x \) as a function of \( y \). Since \( x \geq 0 \), we will consider the positive root when solving for \( x \).

We start with the function \( f(x) = 10 - x^2 \) where \( x \geq 0 \).

To find the inverse function \( f^{-1}(x) \), we set \( y = 10 - x^2 \) and solve for \( x \) in terms of \( y \).

Rearrange the equation to express \( x \) as a function of \( y \): \[ y = 10 - x^2 \implies x^2 = 10 - y \] Taking the square root of both sides, we have: \[ x = \pm \sqrt{10 - y} \]

Since \( x \geq 0 \), we select the positive root: \[ x = \sqrt{10 - y} \]

Final Answer