"As a function of"

How to write functions:

A function is the relation between a set of inputs and a set of possible outputs with the property that each input has only one output.  This means that a function's input and output values are related.

When writing a function such as [f(x)] is the set of possible outputs for every input value which is represented [x]

Taking both of those statements into account, one would be able to find the relationship between many different things and define those relationships as functions.

For example: you would be able to find the relationship between this equilateral triangle's area and "x".  

The first step in doing this is to start out with the equation of what you are finding the relationship of, in this case, it is area, so you would start out with:

Where "b" is the "base" and "h" is the height. 

In this situation we know half of the overall "base"  which is "2x" so the entire base must be "4x". We know have part of our equation:

 Since this is an equilateral triangle we also know that every side is congruent or equal so that means the all sides are "4x".

This bring us to our second step. What we still don't know, is "h" or the "height" and when you are looking at the main goal, which is defining the triangle's Area as a function of "x" you can't have more than one variable (x) in there.  (That means that "h" is not allowed).

To find "h" in terms of "x" we can do one of two things.

1. We could possibly recognize this is a 30,60,90 triangle, making "h"  
2. If we couldn't figure out this is a 30,60,90 triangle, we could do the pythagorean theorem

This means it's time for step 3! We can finish our equation with only one variable so we plug that into our original equation:

Step 4 is to simplify, our first equation can be simplified to:

But wait there's more!

The last, and final step is to convert the equation into a function so we do the following:

Which tells us that "Area" or "A" as a function of "x" is equal to ""

One can write much more than just the Area as a function of "X".  One can write the perimeter of a rectangle as a function of it's area, or the volume of a package as a function of it's base, the possibilities are endless!

Review of steps:
1. Write out equation of what you are trying to find a function of
2. Make sure there is only one variable, and if there is more than one, solve in the terms of what you are looking for.  (if you are looking for A(x) then solve in terms of "x")
3. Plug in the new values for everything into your original equation
4. Simplify!
5. Convert your equation to a function.