Question

determine which of the following relations are functions. select three that apply.
a. {(1,3), (2,3), (3,3), (4,3)}
b. {(3,0), (4,1), (5,2), (4, -1)}
c. {(-2,5), (-1,2), (0,1), (1,2)}
d. {(-3,7), (1,4), (2,9), (5,0)}
e. {(8,1), (8,2), (8,3), (8,4)}

Explanation:

Step1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value). In other words, no two ordered pairs in the relation can have the same x - value with different y - values.

Step2: Analyze Option A

For the relation \(A=\{(1,3),(2,3),(3,3),(4,3)\}\), we check the x - values: 1, 2, 3, 4. Each x - value appears only once. So, each input has exactly one output. Thus, A is a function.

Step3: Analyze Option B

For the relation \(B = \{(3,0),(4,1),(5,2),(4, - 1)\}\), the x - value 4 appears twice, with \(y = 1\) and \(y=-1\). Since the input 4 has two different outputs, B is not a function.

Step4: Analyze Option C

For the relation \(C=\{(-2,5),(-1,2),(0,1),(1,2)\}\), the x - values are - 2, - 1, 0, 1. Each x - value appears only once. So, each input has exactly one output. Thus, C is a function.

Step5: Analyze Option D

For the relation \(D=\{(-3,7),(1,4),(2,9),(5,0)\}\), the x - values are - 3, 1, 2, 5. Each x - value appears only once. So, each input has exactly one output. Thus, D is a function.

Step6: Analyze Option E

For the relation \(E=\{(8,1),(8,2),(8,3),(8,4)\}\), the x - value 8 appears four times, with different y - values. Since the input 8 has four different outputs, E is not a function.

Answer:

A. \(\{(1,3),(2,3),(3,3),(4,3)\}\), C. \(\{(-2,5),(-1,2),(0,1),(1,2)\}\), D. \(\{(-3,7),(1,4),(2,9),(5,0)\}\)