Determine the inverse of \( y=x^{7}+3 \) A \( y=\sqrt[7]{x+3} \) B \( y=\sqrt[7]{x}-3 \) C \( y=\sqrt[7]{x-3} \) D \( y=\frac{x+3}{7} \)

To find the inverse of the function \( y = x^{7} + 3 \), we first need to express \( x \) in terms of \( y \). Start by isolating \( x^{7} \): 1. Subtract 3 from both sides: \[ y - 3 = x^{7} \] 2. Now, take the seventh root of both sides: \[ x = \sqrt[7]{y - 3} \] Thus, the inverse function can be expressed as: \[ y = \sqrt[7]{x - 3} \] Looking at the answer choices, we can see that option C, \( y = \sqrt[7]{x - 3} \), is correct. As for a bit of extra knowledge, did you know that inverse functions essentially "reverse" the action of the original function? This means if you apply the inverse to the result of the original function, you should get back to your starting point! So if you compute \( f^{-1}(f(x)) \), you'll land right back at \( x \) - it's like magic but with math! Additionally, inverse functions are not just crucial in algebra; they show up frequently in calculus, engineering, and even economics. They help us solve equations and understand relationships between variables better. Fun fact: did you know that every time you solve for \( y \) as a function of \( x \), you're essentially looking for a way to navigate through the maze of numbers back to where you started? It's both challenging and rewarding!