Think of a function as a simple machine. You put something in, it does something specific, and one thing comes out. For example, a coffee machine is a function. You put in coffee grounds and water (the input), and you get brewed coffee (the output). You'll always get coffee, not tea or juice. In math, functions work the same way, but with numbers.

A function, usually denoted as $f(x)$, takes an input value, $x$, and produces a single output value, $f(x)$ (read as "f of x"). The set of all possible inputs is called the domain, and the set of all possible outputs is called the range.

The most important rule is that for any given input, there can only be one output. If our coffee machine sometimes produced hot chocolate from the same inputs, it wouldn't be a reliable function.

Functions come in many forms, but a few types show up constantly. Understanding their basic shapes and rules is key.

Linear Functions These are the simplest. They always produce a straight line when graphed. The general form is $f(x) = mx + b$. The value $m$ is the slope (how steep the line is), and $b$ is the y-intercept (where the line crosses the vertical axis).

Quadratic Functions These functions have the form $f(x) = ax^2 + bx + c$. Their graphs are parabolas, the U-shape you might see when you toss a ball in the air. The $ax^2$ term is what makes it a curve instead of a straight line.

Polynomial Functions This is a broader category that includes both linear and quadratic functions. The general form is $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$. They are defined by having variables raised to non-negative integer powers. Their graphs can have various curves and turns.

Now for a core concept in calculus: the limit. A limit describes what a function is doing at a certain point. More specifically, it tells us the value a function approaches as its input gets closer and closer to a particular number.

We write a limit like this:

This is read as "the limit of $f(x)$ as $x$ approaches $c$ is $L$ ". It means that as the input $x$ gets arbitrarily close to the value $c$, the output $f(x)$ gets arbitrarily close to the value $L$.

So how do we find $L$? For many straightforward functions, it's as simple as it looks.

Direct Substitution For most polynomial functions you'll encounter, you can find the limit by simply plugging the value $c$ into the function. Let's find the limit of $f(x) = x^2 + 3$ as $x$ approaches 2.

We just substitute $x=2$ into the expression:

So, the limit is 7. As $x$ gets closer to 2, the function's value gets closer to 7. Easy enough.

When Substitution Fails But what about this function as $x$ approaches 1?

If we try to plug in $x=1$, we get:

This is an "indeterminate form." The function is undefined right at $x=1$. But remember, a limit doesn't care about what happens at the point, only what happens as we get close to it.

We can simplify the expression first. The numerator, $x^2 - 1$, can be factored into $(x-1)(x+1)$. This gives us a new way to write the function:

Now we can cancel the $(x-1)$ terms. This is allowed because we're only concerned with values of $x$ near 1, not equal to 1, so $x-1$ is not zero.

Now we can use direct substitution:

Even though the original function has a hole at $x=1$, its limit as $x$ approaches 1 is 2. This algebraic trick of factoring and canceling is a common way to solve limits that result in $0/0$.

Time to check your understanding.