When you're working with composite functions, it's easy to make some common mistakes that can cause confusion. Here are a few to watch out for:

1. Not Understanding Notation: Sometimes, students mix up $f(g(x))$ with $g(f(x))$. It's important to remember that the order of the functions matters a lot.

2. Ignoring Domain Limitations: If you don’t think about the domain of $g(x)$, you might end up with values that don't work when you try to plug them into $f(x)$. Always check if $g(x)$ is appropriate for $f(x)$.

3. Skipping Simplification Steps: If you skip simplifying expressions or miss steps, you might get the wrong final answer. Make sure to go through each step carefully.

To avoid these mistakes, use clear notation, pay attention to domain limits, and always take time to simplify at every step. Regular practice and being careful can really help you understand composite functions better.