Relations and Functions

Relations and Functions

No one did yesterday's blog entry... Here is some definitions.

"A Relation is when our input (independent/x value) produces an output (dependent/y value). Each input must have at least one output to be a relation."

"A Function is a type of relation where one input will produce exactly one output. Therefore, every function is a relation, but remember every relation is not a function."

If you are ever asked, "Are these functions?" and are presented with a bunch of relations, it's comparable to being presented with a basket of fruit, and having to separate all the apples.
^ Disregard this if that was a horrible comparison. (Attempt to make this post relatable? ☑)

** This blog post will not be showing you the V.L.T. or Vertical Line Test. Sorry, not my job.**

Today's Notes:

Relations and functions can be showed as an equation, graph, or a set of ordered pairs.
Example of an Equation: (y=mx+b)
Example of Ordered Pairs: { (3,2) (4,2) (5,2) }
Example of a Graph:
 

Domain and Range

Domain: the set of all x values that are used as input in a relation.

Range: the set of all y values that are produced as output in a relation.

To write down domain or range you can use "Interval Notation" the simpler way. Or the more complex and widely accepted way, "Set Notation".

Writing Down Domain and Range Interval Notation:

Interval Notation

This is an example graph that we did in class.

To write the domain in Interval Notation for this graph, you simply find the the first and last number on the x axis that this line would sit below or hover over because we have a line between two points, meaning the input could be any number in between these points. If you squint a little, and take a few steps back, you might see that this line(the side with the dot) begins under the -3 on the x axis. This is the first possible input that can be used in the function. The arrow on the other end means that the line continues forever, meaning that the other side of the x axis is extended infinitely. This means the last number on the x axis could be any number above -3, and we'll call this infinity ∞. Therefore to write this in interval form, you'd write:

[-3, ∞)
Congratulations. You did it.
To get your range, you take the line and compare it to it's position along the y axis. Your range is [-2,∞)

But wait, this is the easy way out. The way you should learn this stuff is much more difficult.

(If you come across a graph with points plotted, like the ones in this ordered pair for example, { (3,2) (4,2) (5,2) }

the x values are the first numbers in the pair, and the y values are the ones second.)

Domain: [3,4,5]

Range: [2] or perhaps just 2 is acceptable. I'd mark your page correct either way.

Set Notation

Set Notation is written in this format:
{x/xER}

Translation:

The set of all values of x such that x is an element of the real number system.
*The colour system should be self explanatory*

This means every possible input is part of the real number system.

After this, you write a weird form of interval notation. Using the example above, in the domain, the first number was -3, and the last number was
∞.
Or in other symbols:

x≥-3
Translation:

x is greater than or equal to -3.

All there is left to it, is to tack on the second part, and add it to the first.
{x/xER,x≥-3}
and there is your extremely hard to explain, underwhelming explanation for writing domain.

I won't bother to explain how you get the range for the above example, but here is the Set Notation:

{y/yER,y≥-2}

Apply what you know about domain to range, and there you have it.

Names and or Meanings of the "other symbols"

{ } - Set notation brackets.
[ ] - I call 'em square brackets. They mean "up to and including this number."

( ) - Round brackets? Perhaps? They mean "up to and not including this number." You use these for infinity because you can't quite reach infinity.

Closing

Did I succeed in creating a blog post that was quick, easy to comprehend, and wasn't a wall of text? ☒ Probably not, but I hope it covered everything. So, I know the question we're all wondering here. What if a graph has a line with an arrow on each end? How many numbers will the graph include? Let's ask a mathlete.

(I don't know how terribly relevant this is, but I heard "limit" in class once.)