What does f(x) actually mean?

A great way to understand functions is to think of a factory machine. You put raw materials in, the machine does a specific job, and a finished product comes out. In algebra, the input is your $x$ value, the machine's rule is the equation, and the output is $f(x)$.

If the machine's rule is to double the input, putting in a $3$ means a $6$ comes out. In function notation, we write this as $f(3) = 6$. It compactly tells us "when I put $3$ into machine $f$, I got $6$."

Let us break down the symbols. The letter on the outside, usually $f$, $g$, or $h$, is just the name of the function. We use different letters so we can tell multiple functions apart on the same page.

The variable inside the parentheses, usually $x$, is a placeholder for your input. When you see $f(2)$, it means "take the rule for $f$, and everywhere you see an $x$, replace it with a $2$."

You might wonder why we complicate things by replacing a simple $y$ with $f(x)$. The answer is that function notation gives us a lot more information at a glance.

If you have $y = 2x$ and $y = 3x - 1$, and someone says "evaluate for $x=4$," it can get confusing. If you name them $f(x) = 2x$ and $g(x) = 3x - 1$, you can just say "find $g(4)$." It tells you exactly which equation to use and exactly what to plug in.

The single most common mistake students make is thinking that $f(x)$ means "$f$ multiplied by $x$." In almost all of algebra, two things next to each other mean multiplication. Function notation is the major exception.

The parentheses here do not mean multiply. They are holding the input value. You cannot "divide both sides by $f$" to get $x$ by itself, because $f$ is not a number; it is just a name.