Question

which of the following relations represents a function?
i (1, 0.5), (2, 0.25), (3, 0.125), (4, 0.0625)
ii (-2, 4), (-1, 2), (1, 2), (2, 4)
iii (5, 4), (5, 6), (9, 4), (9, 6)
a i and iii
b i only
c ii only
d i and ii
e i, ii, and iii

Explanation:

Step1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value).

Step2: Analyze Relation I

For the relation \(I:(1, 0.5),(2, 0.25),(3, 0.125),(4, 0.0625)\)
Check the x - values: \(x = 1,2,3,4\). Each x - value appears only once, so each input has exactly one output. So, relation \(I\) is a function.

Step3: Analyze Relation II

For the relation \(II:(- 2,4),(-1,2),(1,2),(2,4)\)
Check the x - values: \(x=-2,-1,1,2\). Each x - value appears only once, so each input has exactly one output. So, relation \(II\) is a function.

Step4: Analyze Relation III

For the relation \(III:(5,4),(5,6),(9,4),(9,6)\)
Check the x - values: \(x = 5\) appears twice (with \(y = 4\) and \(y = 6\)) and \(x=9\) appears twice (with \(y = 4\) and \(y = 6\)). So, there are inputs that have more than one output. Thus, relation \(III\) is not a function.

Answer:

D. I and II