Which of the following is the inverse of the translated quadratic function \( f(x)=x^{2}+15 \) with domain restricted to \( x \geq 0 \) ? Option \#1: \( f^{-1}(x)=\sqrt{x-15} \) Option \#2: \( f^{-1}(x)=\sqrt{x+15} \) Option \#3: \( f^{-1}(x)=\sqrt{x}-10 \) (1 point)

To find the inverse of the function \( f(x) = x^2 + 15 \) with the domain \( x \geq 0 \), we first set \( y = f(x) \) or \( y = x^2 + 15 \). Rearranging this gives \( x^2 = y - 15 \). Since the domain is restricted to \( x \geq 0 \), we take the positive square root, resulting in \( x = \sqrt{y - 15} \). Therefore, the inverse function is \( f^{-1}(x) = \sqrt{x - 15} \). So, Option \#1: \( f^{-1}(x) = \sqrt{x - 15} \) is the correct answer! Interesting fact: Quadratic functions have parabolic shapes, and when their domain is limited, they can have well-defined inverses. This allows us to work with functions that are otherwise not one-to-one! In real-world applications, quadratic functions often model phenomena like projectile motion or the area of a square. By understanding their inverses, we can find missing dimensions or determine how much time something will take to hit the ground based on its launch height!