4. Find the inverse of f(x)=-sqrt (x+2)

To find the inverse of the function \( f(x) = -\sqrt{x+2} \), we need to follow these steps:<br /><br />1. **Express \( y \) in terms of \( x \):**<br /> \[<br /> y = -\sqrt{x+2}<br /> \]<br /><br />2. **Swap \( x \) and \( y \):**<br /> \[<br /> x = -\sqrt{y+2}<br /> \]<br /><br />3. **Solve for \( y \):**<br /><br /> - First, isolate the square root term:<br /> \[<br /> x = -\sqrt{y+2} \implies -x = \sqrt{y+2}<br /> \]<br /><br /> - Square both sides to eliminate the square root:<br /> \[<br /> (-x)^2 = (\sqrt{y+2})^2 \implies x^2 = y + 2<br /> \]<br /><br /> - Finally, solve for \( y \):<br /> \[<br /> y = x^2 - 2<br /> \]<br /><br />4. **Write the inverse function:**<br /> \[<br /> f^{-1}(x) = x^2 - 2<br /> \]<br /><br />So, the inverse of the function \( f(x) = -\sqrt{x+2} \) is:<br />\[<br />f^{-1}(x) = x^2 - 2<br />\]