1. Intro to Functions

2. What is a Function?
A function is a special relationship where each input has a single output. It is often written as "f(x)" where x is the input value.

3. The equation/function notation
What is a Function?
For Example:
FUNCTION
THE RULE
Input
Output x
f(x)
The input value
The output value
The equation/function notation x
f(x)
f(x) = 2x 1 2
f(1) = 2x
2(1) = 2
We replace x with 1 and solve the equation 4
f(2) = 2x
2(2) = 4
replace x with 2 and solve the equation 3 6
f(3) = 2x
2(3) = 6
replace x with 3 and solve the equation
When we write this as a function, it means that for every input x there is a single output for f(x)
THE RULE STATED ABOVE IS TO MULTIPLY THE INPUT BY 2!

4. So What is a Function?
Well a function is like a box with a special rule. You put something in the box and after it goes through the box’s process it produces a unique result. Anything that goes IN the box (the INPUT) is called the DOMAIN. Whatever comes OUT of the box (the OUTPUT) is called the RANGE.

5. For example How do you know it’s a function?
Input
f(x) = 3x + 1
Output x
f(x) 7 2 -2 -5 10 31
How do you know it’s a function?
When each input has exactly one unique output!

6. A function can be represented by a graph
Once a function is represented by ordered pairs or a table of functions, it is easy to plot these points in a graph. Graphs are visual representations of a function on the coordinate plane. Below is an example

7. Graphs are especially useful when determining whether or not a table of values or a set of ordered pairs is actually a function. By plotting the points on the graph, you can now perform a vertical line test on the values. By sweeping a vertical line across the graph you can look to see if any two points occur on the vertical line at the same time. If they do then a function does not exist

8. A function can be represented with mapping.
Mappings are another way to represent functions. Very much like a table in that input values are listed vertically on the left and the output values vertically on the right. The big difference though is that arrows are drawn to show the result of each input. In the illustration below each x is mapped onto it's appropiate f(x) value.

10. Equation v Function Notation
Equations are a textual way to represent functions.. There are many types of equations that represent many types of the functions. For this page, though, we will concrete on linear equations in slope intercept form (y = mx + b). The graphic below illustrates the parts of an equation in slope intercept form.
Other names for input(x) and output(y or f(x)) is domain and range respectively. The domain is the set of all possible inputs and the range is the set of all possible outputs.

11. There are numerous functions in mathematics