Blog 4 - Be the Professor: Domain and Range

Hello everyone,
my name is professor Maragkos and today ill be teaching you what the domain and range of a function is.
To begin with, we need to know the domain and range of a relationship in order to determine whether it is a function or not, and what values this function can have.
So, lets say that we have a few relationships of x and y values.
(2,4), (5,-1), (-1,2), (-4, 6) and (2, 3)
The x-values of the above relationship are its domain, while the y-values its range.
So, the domain of the above relationship is {-4, -1, 2, 5}
and the range {-1, 2, 3, 4,6}
The above set of values does represent a relationship but since there are two sets with the same x-values, (2,4) and (2,3), the relationship does not represent a function. In that case, we can say that the relationship does not pass the horizontal line test, where every x-value must have only one y-value, so it is not a function.
So, in order to determine whether a relationship is a function, we need to look for x-duplicates at the given relationships.

To get deeper into the concept of domain and range, lets say that we are given a fraction, for example 2x+5/x-7
First step is to make sure that the expression is not divided by 0, in other words that x is never 0.
So, taking the denominator and solving for 0
x-7=0 => x=7
We can conclude that the domain of the fraction is all x-values except for x=7
This can be written in many different ways.

First, there is the set builder notation which has the form of {x ε R/ -∞ < 7 < ∞}
And second, there is the Interval notation which has the form of (-∞,7) U (7, ∞)

Determining the range is a little bit trickier, and usually we are asked to sketch or given a graph, to make the process easier.
Lets say the we are given a polynomial function, y= -x^4 + 4 

The domain of this function is all x-values or given in interval notation (-∞, ∞) 

To determine the range of the above function, we need to take a look at the graph

As we can see, the y-values of the function go up to 4 and down to infinity. 

So, the range of the function would be [4, ∞)
or {y ε R/ y>=4}