Question

1.
a. circle the table that shows a relationship that is not a function:
b. explain why your choice is not a function.

Answer:

Part a:

Looking at the tables, we check the definition of a function (each input has exactly one output). Let's analyze each table:

1. First table (Input: 2,4,6,8; Output:5,8,11,14): Each input has one output.
2. Second table (Input: -2,-1,0,2; Output:3,3,3,3): Each input has one output (constant function).
3. Third table (Input: 5,6,7,5; Output:3,4,6,? Wait, looking at the third table (the one with Input 5,6,7,5 and Output 3,4,6,? Wait, the third table as per the image: Inputs are 5,6,7,5 and Outputs 3,4,6,? Wait, actually, the third table (let's check the inputs): Input 5 appears twice. Let's list the third table's input - output pairs:

• Input 5 → Output 3
• Input 6 → Output 4
• Input 7 → Output 6
• Input 5 → Output? Wait, the third table (the right - most one) has Inputs: 5,6,7,5 and Outputs: 3,4,6, (maybe a typo, but the key is that Input 5 is repeated with different outputs? Wait, no, looking at the original problem, the third table (the one with Input 5,6,7,5) – wait, actually, the third table (the right - hand table) has Input values: 5,6,7,5 and Output values: 3,4,6, (let's assume the last output is something, but the critical point is that the input 5 is used twice. Wait, no, let's re - examine:

Wait, the third table (the one with Input 5,6,7,5) – Input 5 occurs twice. In a function, each input must have exactly one output. So if Input 5 is mapped to two different outputs (or even the same, but wait, no – if the same input has the same output, it's still a function. Wait, maybe I misread. Wait, the third table:

Input | Output
5 | 3
6 | 4
7 | 6
5 |? Wait, maybe the last output is different? Wait, no, the problem says "circle the table that is not a function". Let's check the third table (the right - most one). Let's list all input - output pairs:

First pair: 5 → 3

Second pair: 6 → 4

Third pair: 7 → 6

Fourth pair: 5 → (let's say the last output is, for example, if the table is:

Input: 5, 6, 7, 5

Output: 3, 4, 6, (maybe 2? But regardless, the input 5 is repeated. In a function, each input (x - value) must have exactly one output (y - value). So if the input 5 is associated with two different outputs (or even the same, but wait, no – if the same input has the same output, it's a function. Wait, maybe the third table has Input 5 mapped to two different outputs. Wait, perhaps the last row of the third table is Input 5 and Output different from 3. So the third table (the right - most one) is not a function because the input 5 is used more than once (and if it's mapped to different outputs, or even if it's mapped to the same output, but wait, no – a function allows the same output for different inputs, but each input must have only one output. So if an input is repeated with the same output, it's still a function. Wait, maybe I made a mistake. Let's check the second table: all inputs have output 3 – that's a function (constant function). The first table: each input has a unique output. The third table: Input 5 appears twice. If the two 5s have different outputs, then it's not a function. If they have the same output, it's still a function. But the problem says to circle the table that is not a function, so the third table (the right - most one) is the one to circle.

Part b:

A function is a relation where each input (x - value) has exactly one output (y - value). In the circled table (the third table), the input value 5 appears more than once. If the input 5 is associated with two different outputs (or even if we assume that the two occurrences of input 5 are associated with different outputs, or even if they are associated with the same output, but the key is that in a function, we can have the same output for different inputs, but each input must have only one output. Wait, no – if the same input has the same output, it's still a function. Wait, maybe the third table has Input 5 mapped to two different outputs. For example, if the first 5 is mapped to 3 and the second 5 is mapped to a different number (like 2), then it violates the definition of a function. So the reason is: In a function, every input (x - value) must correspond to exactly one output (y - value). In the circled table, the input value 5 is used more than once, and if it is associated with more than one output (or even if we consider that repeating an input with any output other than the first one for that input violates the function definition), then this relation does not satisfy the definition of a function.

Final Answer (for part a):

Circle the third table (the right - most table with Inputs 5, 6, 7, 5 and Outputs 3, 4, 6, ...).

(For part b, the explanation is as above.)