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Steps for Building Functions in Mathematics

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Math functions are relationships between variables that are based upon logical mathematical steps. In this lesson, we will discuss the steps that it takes to build a mathematical function.

What is a Mathematical Function?

'Imagine that Charlie has a shoe factory. Each shoe that Charlie makes requires two square feet of leather. What is a function that would express the relationship between the amount of leather required and the number of shoes produced?'

In mathematics, a function is a relationship between an input, the allowable values that you want to put into the function, and an output, values that come out after the function is applied. It is like a little factory that takes in one kind of thing, applies some sort of procedure to it, then sends out a new thing. For example, you might have a function that you are willing to apply to any real number, where you double that number to produce a result. If you plug a 3 into your function (the input), then doubling it will produce a 6 on the output.


Function Machine
Function Machine


You can name a function anything you like, but a typical name will be a letter (the name of the function, such as f), followed by the variable that will be used to represent the possible input values (such as x). The function name would then be f(x) (called ' f of x '). Sometimes, meaningful words are used for functions, such as 'Earned Income' or 'Resistance Coefficient'.

Steps to Building a Function

Building a mathematical function requires determining a name for the function, defining which values are acceptable as inputs (the domain), defining the process that will be applied to the inputs, testing the possible outputs (range) for invalid results, possibly redefining the domain to eliminate invalid results, and then writing the function to show the relationship. Let's take a look at each of these steps.

Name It

What are you going to call your function? As mentioned above, useful names are usually letters positioned next to a set of parentheses that enclose the variable representing the set of values that the function will be applied to. Typical functions will be f(x), g(y), or possibly f(g(x)), if you want to apply a function to the output of another function. But a function can also be a name, like 'Revenue,' which tells you what the output values are representing, or a combination of names, such as 'Distance Over Time,' which would be a plain English version of some aspect of the relationship between the input and the output.

Define the Input Parameters

The input parameters are a definition for the values that you are going to allow into your function. For example, if you are creating a function that shows Sales as a function of Advertising Budget, you will probably not want to include negative values for the Advertising Budget input, since budgets are generally not negative numbers. The range of possible inputs is called the domain and may be expressed in a list, such as ('x = 1, 2, or 3'), a relationship ('x is greater than 0' or 'x > 0'), or an ordered pair, which matches inputs to outputs, such as (1,3), where 1 is the input and 3 is the output.

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