function or not a function?

Question

hi there i cant seem to find how to work these out i know it is very is easy but how can you tell if it is a function or not??

1) { (x,y): y^2 = x^2 + 4 , xEZ }
2) x -> square root x^3 , x is a positive integer
3) { (1,2) , (1,3) , (2,6) , (3,5) }
4) { (2,1) , (3,1) , (4,1) , (5,1) }

thank you for your help!!

Answer 1

Functions are either:
a) 1 to 1 (i.e. 1 x value for 1 y value), e.g. y = x+2 or
b) many to to 1 (i.e. many x values to 1 y value) e.g. y = x^2

1) { (x,y): y^2 = x^2 + 4 , xEZ } = many to many
=> NOT FUNCTION

2) x -> square root x^3 , x is a positive integer = many to 1
=> FUNCTION

3) { (1,2) , (1,3) , (2,6) , (3,5) }
= 1 x value (ie. 1) has 2 y values
= one to many
=> NOT FUNCTION

4) { (2,1) , (3,1) , (4,1) , (5,1) }
= many x values (ie. 2,3,4) have 1 y value (ie. 1)
=> FUNCTION

Answer 2

1) { (x,y): y^2 = x^2 + 4 , xEZ } Do not understand xEZ, so can't help you with this one.

2) x -> square root x^3 , x is a positive integer. This is a function because there is one value of y for each value of x. This is rater strange notation.

3) { (1,2) , (1,3) , (2,6) , (3,5) }
This is not a function because you have the value x =1 representing two different values of y.

4) { (2,1) , (3,1) , (4,1) , (5,1) }
This is a function no value of x points to more than one value of y. In fact the equation is y =1 which is a horizomtal line parallel to the x-axis. It thus passes the vertical line check

Answer 3

1 is NOT a function because it is the equation for a hyperbola. Hyperbolas fail the vertical line test (and so do circles)

2 I'm not quite sure what the "function" is, so I can't help you here

3 Is NOT a function because it fails the vertical line test. If you place a vertical line at x=1, it intersects the curve at 2 and 3.

4 Is a function. It's a horizontal line.

Answer 4

If you can find two points (x,y1) and (x,y2) with y1 not equal to y2, then it is not a function, Otherwise, it is.

For example, in 3) { (1,2) , (1,3) , (2,6) , (3,5) }, we have the points (1,2) , (1,3), so this is not a function.

Answer 5

it is a function if for every x in the domain , you have one and one only value to y
1) & 3) are not functions but 2) & 4) are functions