Think of a function as a simple machine. You put something in, and you get something out. For any specific input, this machine will always give you the exact same, single output. It's a consistent rule that connects inputs to outputs.

For example, a machine that adds 3 to any number you give it is a function. If you put in 2, you get 5. Every single time you put in 2, you will get 5. You'll never put in 2 and get 8. That's what makes it a function. A single, predictable output for each input.

Instead of writing "a rule that takes a number $x$ and squares it," we use a shorthand called function notation. It looks like this:

Here's how to read it:

• $f$ is the name of the function. We could also call it $g$ or $h$ or even something descriptive like Area.
• $(x)$ holds the input variable. In this case, it's $x$.
• $f(x)$ represents the output of the function $f$ when the input is $x$. We often say "f of x".

If we write $y = f(x)$, it just means that the output value, $y$, is determined by the function $f$ acting on the input $x$.

Let's define our "add 3" function from before. We can write its rule like this:

To find the output for an input of 2, we replace $x$ with 2:

For an input of 10, we get:

Here's a quick look at some inputs and outputs for this function:

Every function has two important sets of values associated with it: its domain and its range.

Think of the domain as all the things you're allowed to put into the function machine. For our simple function $f(x) = x + 3$, we can put in any number we can think of, so its domain is all real numbers.

The range is all the possible results that can come out of the machine. For $f(x) = x + 3$, the range is also all real numbers, because for any number you want as an output, you can find an input that produces it.

However, the domain and range aren't always "all real numbers." Consider the function $g(x) = x^2$.

The domain is still all real numbers, because you can square any number. But what about the range? If you square any number (positive or negative), the result is always positive or zero. You can't get a negative number. So, the range of $g(x) = x^2$ is all non-negative numbers ($y \ge 0$).

Let's review the key terms we've just covered.

Time to check your understanding.

Functions are a fundamental concept in mathematics. Understanding them as a clear rule connecting inputs to unique outputs is the first step to mastering more advanced topics.