Are the following functions inverses? \( \begin{array}{l}f(x)=6 x+3 \\ g(x)=\frac{x}{6}-3 \\ \text { Yes, they are inverses } \\ \text { No, they are not inverses }\end{array} \)

To determine if the functions \( f(x) = 6x + 3 \) and \( g(x) = \frac{x}{6} - 3 \) are inverses, we need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \). Calculating \( f(g(x)) \): \[ f(g(x)) = f\left(\frac{x}{6} - 3\right) = 6\left(\frac{x}{6} - 3\right) + 3 = x - 18 + 3 = x - 15 \] Now calculating \( g(f(x)) \): \[ g(f(x)) = g(6x + 3) = \frac{6x + 3}{6} - 3 = x + \frac{1}{2} - 3 = x - \frac{5}{2} \] Since neither \( f(g(x)) \) nor \( g(f(x)) \) equals \( x \), the functions are not inverses. So the answer is: **No, they are not inverses.** --- To add more fun to this exploration, let's think about their behaviors! Imagine taking a trip in a magical ride represented by \( f(x) \): you zoom ahead 6 times as fast, then get a boost of 3. But then you hop onto \( g(x) \): you slow down to one-sixth the speed and take away 3 meters from your journey. While both journeys sound exciting, they don't quite land you back where you started, which is why they can’t be best buddies in the inverse world! For a deeper dive, you could explore the concept of inverse functions further by examining real-world examples, like how speed and time relate in a car journey. Or check out online resources explaining these concepts in more visual formats. You might find some interactive applets that help visualize how one function flips another around!