When students learn about composite functions in Grade 9 Algebra I, they often run into some common problems. Recognizing these issues can help them understand the topic better and do their work more accurately.

1. Confusing Function Notation:
One big mistake students make is misunderstanding how to read composite functions. For example, if you have two functions, $f(x)$ and $g(x)$, the composite function is shown as $f(g(x))$. Some students misread this as $g(f(x))$, which gives wrong answers. It’s really important to remember the order of the functions when you use them.

2. Ignoring the Domain:
Another issue is forgetting about the domain of the composite function. This means the output of $g(x)$ must fit within the input range of $f(x)$. If it doesn’t, you won't be able to create the composite function. To fix this, students should first check the domains of both functions before combining them.

3. Forgetting to Fully Evaluate:
Sometimes students don’t finish all the steps when calculating composite functions. For example, if they need to find $f(g(2))$, they might successfully calculate $g(2)$ but then forget to put that answer into $f(x)$. This can lead to wrong results. To avoid this, it helps to write down each step clearly and finish one function completely before starting the next.

4. Overlooking Function Types:
Students can also forget that different types of functions can change how the composite function works. For instance, if $g(x)$ is a curve (quadratic) and $f(x)$ is a straight line (linear), the result might not act as expected. Getting to know the different types of functions can make it easier to guess what will happen.

5. Misplacing Parentheses:
Finally, using parentheses incorrectly can cause major mistakes. For example, $f(g(x+1))$ is not the same as $f(g(x)) + 1$. It’s very important to use parentheses correctly and pay attention to the notation to avoid confusion.

In conclusion, while composite functions can be tricky because of these common mistakes, students can improve their understanding by focusing on function notation, domain knowledge, step-by-step evaluation, and correct use of parentheses. Regular practice can help them build their skills and make fewer errors.