Questions: Find the inverse of the function f(x)=-3x^2.

Solution Steps

To find the inverse of the function \( f(x) = -3x^2 \), we need to follow these steps:

1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \).
3. Solve for \( y \) in terms of \( x \).
4. Replace \( y \) with \( f^{-1}(x) \).

We start with the function given by \( f(x) = -3x^2 \).

We set \( y = -3x^2 \).

Next, we swap the variables to get \( x = -3y^2 \).

To find the inverse, we solve for \( y \): \[ -3y^2 = x \implies y^2 = -\frac{x}{3} \implies y = \pm \sqrt{-\frac{x}{3}} \] However, since the original function \( f(x) = -3x^2 \) is not one-to-one over all real numbers, we restrict the domain to \( x \leq 0 \) to ensure the inverse is a function. Thus, we take the negative root: \[ y = -\sqrt{-\frac{x}{3}} \]

Final Answer