## 1.3 Piecewise Functions

Vocabulary words: **2**

A **piecewise function** is the combination of multiple functions, with each covering a different part of the overall function’s domain.

\[f(x)=\begin{cases}
8x && 0 \le x \le 40 \\
12x-160 && x > 40
\end{cases}\]

The domain of the entire function is $[0,\infty)$, but from $[0,40]$ it follows the function $8x$. Beyond that it takes on $12x-60$.

There is not much to these besides understanding how to read them. They are just two (or more) functions graphed together.

### Step Functions

A **step function** is a type of piecewise function that’s made up constant functions, which makes the resulting graph look like steps. Two examples of this you can try out in Desmos are the floor() and ceil() functions. Link here.

The first takes its input and rounds it down. So, for all $1\le x < 2$, the result is 1. The ceil function is the opposite, everything is rounded up. $1 < x \le 2$ yields 2.

### Piecewise Functions on Desmos

Here’s a demo, but what you need to do it put each part on a seperate line, and in braces, put the domain followed by a colon and then the function.

So our piecewise at the top of the lesson looks like this

\[\begin{align*}
y &= \left\{0\le x \le 40:8x\right\} \\
y &= \left\{x>40:12x-160\right\}
\end{align*}\]