Find the inverse of g left parenthesis x right parenthesis equals negative x squared plus 3 for xgreater than or equals0.

Question

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Part 1
Find the inverse of g.
g Superscript negative 1 Baseline left parenthesis x right parenthesisequals
  
enter your response here  ​(Simplify your​ answer.)

Answer 1

To find the inverse of the function \( g(x) = -x^2 + 3 \) for \( x \geq 0 \), follow these steps:

1. Replace \( g(x) \) with \( y \): \[ y = -x^2 + 3 \]

2. Swap \( x \) and \( y \) to find the inverse: \[ x = -y^2 + 3 \]

3. Solve for \( y \): \[ -y^2 = x - 3 \ y^2 = 3 - x \ y = \sqrt{3 - x} \quad (\text{since } x \leq 3 \text{ and } y \geq 0) \]

4. Write the inverse function: \[ g^{-1}(x) = \sqrt{3 - x} \]

Thus, the inverse of the function \( g(x) = -x^2 + 3 \) for \( x \geq 0 \) is: \[ g^{-1}(x) = \sqrt{3 - x} \]