When you start to learn about functions in pre-calculus, the notation $f(x)$ might look a bit confusing at first. But don’t worry! Once you understand it, it becomes a helpful way to tell different types of functions apart.

1. What is $f(x)$?
$f(x)$ is just a way to show a function. Think of it like this: “Let’s see what the function $f$ gives us when we put in $x$.” For example, if we have the function $f(x) = 2x + 3$, it means that when you choose a value for $x$, you will get a specific answer. If you put in $x = 2$, then $f(2) = 2(2) + 3 = 7$.

2. Linear vs. Non-linear Functions:
You can start by telling the difference between linear and non-linear functions:

• Linear functions look like this: $f(x) = mx + b$. They make straight lines on a graph and change at a steady pace (this is called the slope $m$).
• Non-linear functions, like quadratic ($f(x) = ax^2 + bx + c$) or cubic ($f(x) = ax^3 + bx^2 + cx + d$), create curves and do not change at the same rate.

3. Finding the Type of Function:
You can usually figure out what type of function it is by looking at how $f(x)$ is written. Here are some simple tips:

• If $f(x)$ only has variables raised to the first power and some numbers, it's linear.
• If $f(x)$ has variables with powers higher than 1, like $x^2$, then you might be looking at quadratic or other polynomial functions.

4. Constant and Piecewise Functions:
It’s also important to know about constant functions, which just show up as horizontal lines, where $f(x) = c$ (a number). Then there are piecewise functions, which have different rules for different values of $x$. For example, you might see something like this:
$f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ 2x & \text{if } x \geq 0 \end{cases}$.

Learning to understand $f(x)$ helps you work with functions in a smarter way. As you keep learning, remember that this notation is important, not just in pre-calculus, but in harder math topics too!