Establish which of the following options is the inverse of the quadratic function f(x)=3x^2−1

Question

with a domain restricted to [0,∞)
.(1 point)
Responses

Answer 1

To find the inverse of the quadratic function \( f(x) = 3x^2 - 1 \) with a restricted domain of \([0, \infty)\), we can follow these steps:

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

2. Swap \( x \) and \( y \): \[ x = 3y^2 - 1 \]

3. Solve for \( y \): \[ x + 1 = 3y^2 \] \[ y^2 = \frac{x + 1}{3} \] \[ y = \sqrt{\frac{x + 1}{3}} \] (Since the domain is restricted to \([0, \infty)\), we only take the positive square root.)

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

Thus, the inverse of the quadratic function \( f(x) = 3x^2 - 1 \) with domain \([0, \infty)\) is: \[ f^{-1}(x) = \sqrt{\frac{x + 1}{3}} \]

Please select this option if it is among the responses provided.