There are many things parts that go into accurately describing what a function is, what it does, and how to tell if a set of data is, or is not, a function. Throughout this site, I will attempt to accurately describe these things in a manner that is suitable for someone who has no clue what a function is, or how it works.
First off you need to know what exactly a function is.
A function is a set of points, each point containing a domain (x value), and a range (y value).
A function must be pertaining to a set of points on a graph, or from an equation
A function has only one input (x value), for each output (y value)
A function cannot contain multiple y values for one x value
A function can contain multiple x values for one y value
Here is an example of a function and a non-function
Every function contains a parent function (original function) from which is transform (change the function {domain and/or range} around).
f(x)=x Linear
f(x)=x² Quadratic
f(x)=x³ Cubic
f(x)=b^x Exponential
f(x)=1/x Piecewise Function
f(x)=int(x)=[x] Greatest Integer
f(x)=/x/ Absolute Value
f(x)= √x Square root
f(x)=³√x Cube root
f(x)=logb(x) Log
f(x)=1/x² Piecewise Function Squared
f(x)=c Constant Function
Here is an example in graphical form
There are 3 different types of functions, and 4 different criterias of functions
Surjective- The domain(x values) are mapped onto the codomain(y values), so that all of the range values are used up
Injective- The function is one-to-one, meaning that no x values or y values are used twice
Bijective- The function is both Surjective and Injective
Even- When you make x negative in f(x){f(-x)}, then the function becomes negative {-f(x)}
Odd- when you make x negative in f(x){f(-x)}, then the function remains the same {f(x)}
Neither- when you make x negative in f(x){f(-x)}, then the function neither odd, nor even{f(x)=x²+x;f(-x)=x²-x}
Both Even and Odd- When you make x negative in f(x){f(-x)}, the function remains the same and changes{f(x)=0;f(-x)=0}
Inverse Functions
If a function is an inverse, it is injective, or one-to-one
A function that is one-to-one is said to be invertible
If a function is invertible, the composite of the function is equal to x
y=f^-1(f(y))=f^-1(x)
The graph of the function is symmetrical about the line y=x
The domain and range values of an equation and its inverse are swapped
So if f(9)=8 then g(8)=9
Quiz 1: put your knowledge of functions into use
Question 1
What is a function?
A function contains multiple y values for one x value
A function contains only one point on a graph
A function is a set of points, with each point containing an x value and a y value
Question 2
What are the three different types of functions?
Injective, Surjective, and Bilateral
Injective, Surjective, and Bijective
Superman, Batman, and Antman
Question 3
What pertains to an inverse function?
A function that is one-to-one is said to be invertible
A function contains only one x value for every y value
A function is a set of points, with each point containing an x value and a y value
Here is a link to my second page about functions: Functions: Part 2.
