Question

determine the inverse of ( f(x) = (x + 2)^2 - 3 ).

1. ( f^{-1}(x) = -2 + sqrt{x + 3} )
2. ( f^{-1}(x) = -3 + sqrt{x + 2} )
3. ( f^{-1}(x) = -2 + sqrt{x + 3} )
4. ( f^{-1}(x) = -3 + sqrt{x + 2} )

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x) = (x + 2)^2 - 3 \). Replace \( f(x) \) with \( y \), so we have \( y=(x + 2)^2 - 3 \).

Step2: Swap \( x \) and \( y \)

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x=(y + 2)^2 - 3 \).

Step3: Solve for \( y \)

First, add 3 to both sides of the equation: \( x + 3=(y + 2)^2 \).
Then, take the square root of both sides. Since we are dealing with the inverse of a quadratic function (which is a parabola opening upwards, and we assume the principal square root for the inverse function in the context of the given options), we have \( \sqrt{x + 3}=y + 2 \) (we consider the positive square root as the original function \( f(x)=(x + 2)^2-3 \) has a vertex at \( (-2,-3) \) and is increasing for \( x\geq - 2 \), so its inverse will be defined for \( x\geq - 3 \) and increasing).
Next, subtract 2 from both sides to solve for \( y \): \( y=-2+\sqrt{x + 3} \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

We get \( f^{-1}(x)=-2+\sqrt{x + 3} \).

Answer:

The inverse function \( f^{-1}(x)=-2+\sqrt{x + 3} \), which corresponds to option A (assuming option A is \( f^{-1}(x)=-2+\sqrt{x + 3} \) as per the given problem's option layout). If we follow the options as presented (where the first option is \( f^{-1}(x)=-2+\sqrt{x + 3} \)), the answer is the option with \( f^{-1}(x)=-2+\sqrt{x + 3} \), which is the first option (let's assume the options are labeled as 1,2,3,4 and the first one is \( f^{-1}(x)=-2+\sqrt{x + 3} \)). So the answer is the function \( f^{-1}(x)=-2+\sqrt{x + 3} \) (matching the first option in the given set of inverse function options).