Question

3 a relation is represented by the table of values shown. complete the table so that it represents a function. input -10 -2 1 5 output 1 3 4 5

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). So, we need to choose an input value (the missing one in the "Input" column) that is not already present in the existing input values (- 10, - 2, 1, 5).

Step2: Determine the missing input

The existing input values are - 10, - 2, 1, 5. We can choose any value that is not in this set. Let's pick a value, for example, 0 (we could also choose other values like 2, 3, 6, etc.). Let's verify: if the input is 0, then the output is 3. Now, check all inputs: - 10 has output 1, - 2 has output (let's say we can leave the output for - 2 as is, but wait, the output for - 2 is also missing? Wait, the table has input - 10 (output 1), - 2 (output empty), then a missing input (output 3), 1 (output 4), 5 (output 5). Wait, maybe I misread. Let's re - examine the table:

Input column: - 10, - 2, [missing], 1, 5

Output column: 1, [missing], 3, 4, 5

First, for the input - 2, the output can be any value, but the key is that the missing input (let's call it x) must be a value not equal to - 10, - 2, 1, 5. And the output for - 2 can be any value, but let's first handle the missing input.

Let's find a value for the missing input (the third input) such that it is unique. Let's choose x = 2 (or any other number not in {- 10, - 2, 1, 5}).

Now, for the input - 2, the output can be any value, but let's make sure that for each input, there is one output. Let's assume we can fill the output for - 2 as, say, 2 (it can be any value, as long as the input - 2 has one output). Wait, maybe the problem is that both the output for - 2 and the input for output 3 are missing. Wait, the original table:

Input: - 10, - 2, [blank], 1, 5

Output: 1, [blank], 3, 4, 5

So we have two blanks: one in the input (third row) and one in the output (second row, input - 2).

Wait, let's correct the approach.

First, the input values must be unique. So the missing input (let's call it x) must not be - 10, - 2, 1, 5.

The output for input - 2 can be any value, but let's first find the missing input.

Let's list the existing inputs: - 10, - 2, 1, 5. So the missing input (let's say the third input) can be, for example, 3. Then, the output for that input is 3.

Now, for the input - 2, the output can be any value, let's say 0 (or any other value). Let's check the function definition:

- Input - 10: output 1 (unique)
- Input - 2: output 0 (unique)
- Input 3: output 3 (unique)
- Input 1: output 4 (unique)
- Input 5: output 5 (unique)

So, to complete the table:

For the input - 2, we can choose an output (let's pick a number, say 2), and for the missing input (the third input), we can choose a number not in {- 10, - 2, 1, 5}, say 3.

Wait, maybe the problem is that the output for - 2 is also missing. Let's re - express the table:

| Input | - 10 | - 2 |? | 1 | 5 |
|-------|------|-----|----|----|----|
| Output | 1 |? | 3 | 4 | 5 |

So we have two blanks: one input (let's call it x) and one output (for input - 2, let's call it y).

First, the input x must be different from - 10, - 2, 1, 5. Let's choose x = 0.

Then, the output for input - 2 (y) can be any value, let's choose y = 2.

Now, check the function:

- x=-10, y = 1 (unique)
- x=-2, y = 2 (unique)
- x = 0, y = 3 (unique)
- x = 1, y = 4 (unique)
- x = 5, y = 5 (unique)

This satisfies the definition of a function.

Alternatively, if we assume that the output for - 2 is a typo and maybe the user meant that only the input is missing (the third input), let's re - check.

Wait, the original problem says "Complete the table so that it represents a function". So we need to fill both the missing input and the missing output for - 2? Wait, no, looking at the table:

Input: - 10 (output 1), - 2 (output empty), then a cell with empty input (output 3), 1 (output 4), 5 (output 5).

So there are two missing cells: one in the input (the third input) and one in the output (the output for - 2).

First, for the input (third input): must be unique (not - 10, - 2, 1, 5). Let's pick 3.

For the output (output for - 2): can be any value, let's pick 2.

Now the table becomes:

| Input | - 10 | - 2 | 3 | 1 | 5 |
|-------|------|-----|----|----|----|
| Output | 1 | 2 | 3 | 4 | 5 |

Now, check each input:

- - 10: output 1 (unique)
- - 2: output 2 (unique)
- 3: output 3 (unique)
- 1: output 4 (unique)
- 5: output 5 (unique)

This is a function.

Answer:

For the input (third column): we can choose a value not in {- 10, - 2, 1, 5}, e.g., 3. For the output (second column, input - 2): we can choose a value, e.g., 2. So the completed table (one possible completion) is:

Input	- 10	- 2	3	1	5

(Note: The output for - 2 and the input for output 3 can have other valid values as long as the function definition is satisfied. For example, input for output 3 could be 0, and output for - 2 could be 6, etc.)