when do you use arbitrary in describing domain and range of a graph or a table?

Question

my teacher said that in a question I was supposed to graph with any real number and then give the domain and range of that equation I wrote domain: >x> or domain: (,,,,)  he said that the domain and range is arbitrary so I am confused that when you are supposed to say its arbitrary and when you are not.

Answer 1

For a specific function, or graph, or table of points, the domain and range are NOT ARBITRARY.

 

Functions, or graphs, or tables of points, are Relations. A Relation is a set of Ordered Pairs. This is an Ordered Pair:  (x, y); the order matters; x first, then y.

 

The Domain of a Relation is the set of all the x-values in the Relation's set of Ordered Pairs.

 

The Range of a Relation is the set of all the y-values in the Relation's set of Ordered Pairs.

 

Let T = { (-3, 4), (-2, -1), (4, -3), (-2, 2) } be a Relation.

The Domain of T = { -3, -2, 4 } and the Range of T = { -3, -1, 2, 4 }.

 

A Function is a Relation where ordered pairs with the same x-value have the same y-value. If a Relation has two ordered pairs with the same x-values and different y-values, then the Relation is not a Function.

 

P = { (x,y) | y = x^2 + 1 } is a Function with Domain = { x | x is any Real Number } and Range = { y | 1 <= y }.

 

Q = { (x,y) | x = y^2 + 1 } is NOT a Function because y = +- sqrt(x - 1); its Domain = { x | 1 < x } and its Range = { y | y is any Real Number }. Q is the Inverse of P.

 

Answer 2

X: Domain of independent variable (typically.  It's just a label though...could be Q or P or anything)

Y: Range of Dependent variable (typically....same comments)

 

The domain of X is typically driven by the FUNCTION.  Some functions will provide infinities if certain X are used  

 

example 1:

 

Y=mX+B  (Straight line)

 

Domain=<-infinity, infinity>

Range= <-infinity, infinity>

 

example 2: (limited Range)

 

Y=X²  (parabola)

 

Domain=<-infinity, infinity>

Range=<0,infinity>     (Function doesn't output any negative numbers)

 

example 3: (limited Domain and Range))

 

Y=-sqrt(-2x+3)

 

Domain= (-infinity, 1.5)   (Can't take sqrt of negative number)

Range= (-infinity, 0)  (Y never takes a value >0)