To find the inverse of the function [tex]\( f(x) = -\frac{1}{2} \sqrt{x+3} \)[/tex] with [tex]\( x \geq -3 \)[/tex], follow these steps:
1. Start by setting [tex]\( y = f(x) \)[/tex]:
[tex]\[
y = -\frac{1}{2} \sqrt{x+3}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Start by isolating the square root term:
[tex]\[
y = -\frac{1}{2} \sqrt{x+3}
\][/tex]
Multiply both sides by -2 to get rid of the fraction and the negative sign:
[tex]\[
-2y = \sqrt{x+3}
\][/tex]
3. Square both sides to eliminate the square root:
[tex]\[
(-2y)^2 = x + 3
\][/tex]
[tex]\[
4y^2 = x + 3
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4y^2 - 3
\][/tex]
5. Now, express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = 4x^2 - 3
\][/tex]
6. To determine the domain of the inverse function, consider the range of the original function [tex]\( f(x) = -\frac{1}{2} \sqrt{x+3} \)[/tex], which is [tex]\( y \leq 0 \)[/tex] because the function is always non-positive. Therefore, the domain of the inverse function is [tex]\( x \leq 0 \)[/tex].
Summarizing:
- The inverse function is [tex]\( f^{-1}(x) = 4x^2 - 3 \)[/tex]
- The domain of the inverse function is [tex]\( x \leq 0 \)[/tex]
Thus, the correct answer is:
[tex]\[
f^{-1}(x) = 4x^2 - 3 \quad \text{, for } \quad x \leq 0
\][/tex]
