What are Functions?

What are Functions? - How it Works - Video

Function Machine

Function Machine:

Why do we call it the function machine? Any machine you put something into it and you get something out like a carpenter. A carpenter uses pieces of oak and creates a table.

Functions are made up of sets. In the English language the word that has the most definitions is the word set. In math sets are basically a collections of items. It doesn't have to be numbers. It can a collection shirts, letters, days of the week, or ... Whatever you can come up with?

With sets you have two types infinite or finite. An example of finite would be the days of week, {Sunday, Monday, ... , Saturday}. An example of infinite would be the integers, {..., -3, -2, -1, 0, 1, 2, 3, ...} In the first set the ellipse means there are a finite number of elements or members in the set. In the second set the ellipses means that the numbers go on forever or are infinite.

The first set or our yellow circle has several different names including input values, the independent variables, the domain, or the x-values.

The second set or our green circle has several different names, as well, including output values, the dependent variables, the range, or the y-values.

Mapping

Mapping:

The definition of a function is there is one output for every input.

On the left with the letters and the symbols, x, y, and z go to a different symbol. So, it is function.

On the right, we have two inputs of 1, but each 1 has a different output. So, it is not a function.

If we were to graph each one, the one of the right would not pass the vertical line test, which is discussed in the next few sections.

Input - Output - Ordered Pair

Input- Output - Ordered Pair:

We have the function y = 2x + 1. We have created an X-Y table for input values, the x-values, and the output values, the y-values. The great thing is that we can combine those values, to create our ordered pair.

Once we have at least two ordered pairs, we can graph our function on the X-Y grid.

Vertical Line Test

Vertical Line Test:

How can we tell if the graph is a function or not? Well, we can use the vertical line test. You can use your ruler and place it on the graph vertically. If the ruler is touching the graph more than once at any point when you left or right, then the graph is not a function.

The reason for that is functions have a one to one relationship. That means for input value, there is only one output value. Or for every value of x, there is only one value of y.

Vertical Line Test - Example

Vertical Line Test - Example:

Here we have the equation y = x². To make the X-Y table easier to substitute, let's solve for y => y = ± sqrt(x).

There is only input value that gives one output value, (0, 0). Every other input gives two output values. If we take a look the graph, the red line passes through (4, 2) and (4, -2). So how do we know which output value we want if we say our input value is 4? Since we can't be sure, this graph is not a function.

Function Notation

Function Notation:

F(x) = y, we can read that as a function of x equals y or f of x equals of y.

Since f(x) = y or y = f(x), we can substitute f(x) for very y we see. One of the examples above is y = 2x + 1. After we substitute we have f(x) = 2x + 1.

You might different letters than f and that is because there might be more than equation in the question so it used to differentiate the equations.

Function Notation Evaluating

Function Notation Evaluating:

When evaluating a function, it is almost the same before except we start with function(input) = output. Remember our input value is our x-value. So we substitute the value for every x we see in the equation.

The main difference is that at the end we know the input value when we look at the last line.

Live Worksheet

Teacher - Edpuzzle Link