Understanding how functions work is really important when we put them together. Here’s why:

1. Behavior Analysis:

   • It helps to know if a function is going up, down, or staying the same. This way, we can guess what the new combined function will do.

2. Domain Restrictions:

   • Each function can only work with certain values. For example, the function $f(x) = \sqrt{x}$ only accepts values where $x$ is 0 or more. When we combine functions, we must follow these rules.

3. Operations:

   • Addition: When we add two functions, like $h(x) = f(x) + g(x)$, the new function will give us the biggest result from either $f(x)$ or $g(x)$.

   • Multiplication: When we multiply functions, like $h(x) = f(x) \cdot g(x)$, we might find new points where the function equals zero.

4. Inverse Functions:

   • Knowing about inverse functions helps us figure out when a combined function can still go back to the original.

By understanding these ideas, students can work with functions more easily and accurately.