### Definition of a Function

Functions are nothing more than equations where you can plug in an input variable `\(x\)`

and get an output variable `\(y\)`

in return.
```
\[
x \xrightarrow{\text{input}}\boxed{function}\xrightarrow{\text{output}} y
\]
```

Functions describe the relations, causalities, and changes between one variable (*the input*) and the next (*the output*). In function notation it is expressed as:
```
\[
\Large y = f(x)
\]
```

### Examples

*These examples are taken from The Manga Guide to Calculus (highly recommended)*

**An example of Causality**

The frequency of a cricket's chirp `\(y\)`

is determined by temperature `\(x\)`

with the function `\(y = 7x-30\)`

.
Given `\(x\)`

is `\(27^\circ C\)`

:

```
\(\begin{align}
y &= 7x - 30 \\
&= 7(27) - 30 \\
y &= 159 \mbox{ chirps per minute}
\end{align}\)
```

**An example of Changes**

The speed of sound `\(y\)`

in meters per second (`\(m/s\)`

) changes in relation to the temperature `\(x^\circ C\)`

.

```
\(\begin{align}
y &= v(x) \\
&= 0.6x + 331
\end{align}\)
```

```
\[\begin{array}{lll|lll}
x & = & 15^\circ C & x & = & -5^\circ C \\
y & = & v(15) & y & = & v(-5) \\
& = & 0.6(15) + 331 && = & 0.6(-5) + 331 \\
& = & 340 \mbox{m/s} && = & 328 \mbox{m/s} \\
\end{array}\]
```

**An example of Relations**

Conversion between `\(x^\circ\)`

Fahrenheit to `\(y^\circ\)`

Celsius.

```
\[\begin {array}{lll|lll}
y & = & f(x) & x & = & 50^\circ F\\
& = & \frac 5 9 (x - 32) && = &\frac 5 9 (50 - 32) \\
&&&& = & 10^\circ C
\end{array}\]
```

### As an aside...

** Composition of functions** is the combination of two or more functions.

```
\[
x \rightarrow \boxed{f} \rightarrow f(x) \rightarrow \boxed{g} \rightarrow g(f(x))
\]
```

In computer science, functions that can be passed to other functions as input (*arguments*) are called **first class functions**.

### Graphing Functions

Graphing functions is quite simple, you plug in any variable `\(x\)`

into the function and take the result `\(y\)`

and use them as your `\(x, y\)`

coordinates.

`\(f(x) = 2x - 1\)`

```
\[\begin{array}{c|c|c}
x & 2x - 1 & (x,y) \\
\hline
1 & 1 & (1,1) \\
2 & 3 & (2,3) \\
3 & 5 & (3,5)
\end{array}\]
```

You might ask what the purpose of graphing a function is... Graphs are visual representations of data and the effect that they have on each other (sound familiar? that's what a function is supposed to show). These relations are easier to understand when graphed since humans are primarily visual creatures. Graphs are a perfect way to see and predict trends; it is such an effective way that even very young children can comprehend what graphs are saying.