Question

1. three relations are shown below.

i. {(0,1), (1,2), (2,3), (3,4)}
ii. image of a mapping diagram with top oval containing 3,4,5,6 and bottom oval containing 3,4,5,6, each element mapping to itself
iii. image of a graph with points and open/closed circles
which relations represent a function?
a) i, ii, and iii
b) i and ii, only
c) i and iii, only
d) ii and iii, only

Explanation:

Step1: Recall Function Definition

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

Step2: Analyze Relation I

Relation I: \(\{(0,1),(1,2),(2,3),(3,4)\}\). Each input (0, 1, 2, 3) has a unique output. So, it is a function.

Step3: Analyze Relation II

Relation II: A mapping from the set \(\{3,4,5,6\}\) to \(\{3,4,5,6\}\) where each element maps to itself. Each input has exactly one output. So, it is a function.

Step4: Analyze Relation III

Relation III: From the graph, we check the vertical line test. For any vertical line, it intersects the graph at most once. Each x - value (input) has exactly one y - value (output). So, it is a function.

Answer:

a) I, II, and III