Which of the following is the inverse of the function \( f(x)=3 x^{2} \) with a domain restricted to \( x \geq 0 \) ? Option \#1: \( f^{-1}(x)=\sqrt{\frac{x}{3}} \) Option \#2: \( f^{-1}(x)=\sqrt{3 x} \) Option \#3: \( f^{-1}(x)=\sqrt[3]{x} \) (1 point)

To find the inverse of the function \( f(x) = 3x^2 \) with the domain restricted to \( x \geq 0 \), we need to solve for \( x \) in terms of \( y \): 1. Start with \( y = 3x^2 \). 2. Rearranging gives \( x^2 = \frac{y}{3} \). 3. Taking the square root (and noting \( x \geq 0 \)), we get \( x = \sqrt{\frac{y}{3}} \). Thus, the correct inverse function is \( f^{-1}(x) = \sqrt{\frac{x}{3}} \), which corresponds to Option #1. Now, let's add some fun facts! Did you know that the concept of function inverses dates back to the ancient Greeks? They laid down the foundations of geometry, which later influenced mathematics to explore relationships like these in greater depth. The relentless pursuit to understand these relationships has driven advancements over centuries! When working with inverse functions, a common mistake is forgetting the restriction on the domain. Remember, the original function \( f(x) = 3x^2 \) is only defined for \( x \geq 0 \). If you mistakenly try to take the square root of negative numbers, you’ll end up in the realm of complex numbers, which is not what we want here! Always pay attention to the function's original domain to avoid those hiccups!