# Functions.

What are functions? What does $f(x)$ mean?
All will be answered below.

We're already familiar with the forms $y = mx + b$, $y = ax^2+bx+c$, etc.
Function notation (the fancy $f$ ) is just another way to represent this.

Let's take the graph of $y = 2x + 1$:


This is just a basic line that extends in both directions infinitely.

Let's take a closer look at the individual points.

What's the value of the $y$ point on the line when $x = 1$?
We can just substitute that into our equation:

$$
\begin{equation}
\begin{split}
2x + 1 &= y \\
2(1) + 1 &= y \\
2 + 1 &= y \\
3 &= y \\
\end{split}
\end{equation}
$$

Now if we look back at our graph, and we take a look at what the y-value is for the coordinate on the line where $x = 1$:


We can see that $y = 3$, just like we found.

Function notation is just a fancier way to show what we just did.

Let's use the fancy function notation for our line here:

$$
f(x) = 2x + 1
$$

This is read as "f of x is equal to two (times) x plus one".
What this is essentially saying is: "What is the value of y when x is <number> (on this line)?"

So if we had the problem:

$$
\text{Given: } f(x) = 2x + 1 \\
\text{Find: }f(1)
$$

This problem is asking us: "What is the value of y when x is 1?"

$$
\begin{equation}
\begin{split}
f(1) &= 2(1) + 1 \\
f(1) &= 2 + 1 \\
f(1) &= 3 \\
\end{split}
\end{equation}
$$

When $x = 1$, $y = 3$.