## Functions

### 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.