Question 22 Consider the function \( f(x)=\sqrt{x+3} \) Find the formula for the inverse function. \( f^{-1}(x)=\frac{1}{\sqrt{x-3}} \) \( f^{-1}(x)=x^{2}-3 \) \( f^{-1}(x)=x^{2}+3 \) \( f^{-1}(x)=\frac{1}{\sqrt{x+3}} \)

To find the inverse function of \( f(x)=\sqrt{x+3} \), start by switching \( x \) and \( y \) (where \( y = f(x) \)). This gives us \( x = \sqrt{y+3} \). Next, square both sides to eliminate the square root, resulting in \( x^2 = y + 3 \). Finally, solve for \( y \): \( y = x^2 - 3 \). Hence, the inverse function is \( f^{-1}(x) = x^2 - 3 \). Now, when working with inverse functions, it's crucial to ensure the ranges and domains are appropriate. A common mistake is forgetting to restrict the domain of the original function. Since \( f(x) = \sqrt{x+3} \) only takes values where \( x+3 \geq 0 \) (or \( x \geq -3 \)), the inverse function must respect this, hence being valid for \( x \geq -3 \) in the original function.