Functions, Domain, Range, & Function Notation

Functions

 

Everything is a relation; the question is if it is a function. Any set of ordered pairs is a relation. However for it to be a function, no 2 x-values may share 1 y-value.

 

I.e.

 

Set 1: (1,2), (2,3), (3,4), (4,5) -> is a Function

 

Set 2: (1,10), (2,1), (3,2), (4,3) -> is not a Function (Point 1 and 2 share the same y-value)

 

This leads to the vertical line test (VLT). The vertical line test is passing a vertical line through all points. If at one point there are 2 points directly on top of each other, then the relation isn’t a function

 

Domain and Range

 

The set of 1st elements in a relation are the Domain. The set of 2nd elements in a relation are the Range.

 

Relation: (3,7), (2,4), (-3,5), (3,6)

 

Domain (D): (-3, 2, 3) ** In order of least to greatest

Range (R): (4, 5, 6, 7)

 

If the Domain or Range are infinite, it is written as: XER or YER -> X belongs to all real numbers XER or Y belongs to all real numbers YER

 

Function Notation

 

Let y=5x-7

 

Y depends on X. Y is a function.

 

Therefore, f(x)=5x-7 -> Function Notation

 

Function Notation is read as “f of x” or “f at x”

 

A function is a machine. When you input an X you output a Y.


Example:

f(x) = x^2 +7x - 1

f(0) = 0^2 + 6(0) - 1

f(0) = -1

 

 

f(-2) = (-2)^2 + (6)(-2) -1

f(-2) = 4 -12 -1

f(-2) = -9