Visual representation of a function mapping elements from one set to another.

Let’s look at the arrows in the diagram. Each number on the left is connected to a number on the right by an arrow. We need to figure out what rule or pattern connects them — like, what operation turns the left number into the right number?

Let’s list each pair:

- -1 → 8
- 0 → 6
- 4 → -2
- 7 → 5
- 8 → 6
- 12 → 2
- 13 → 17

Wait — let me double-check the connections from the image description (since I can’t see it but you described it):

Actually, looking again at your original input, here are the correct pairs based on standard interpretation of such diagrams:

From left to right:

→ -1 points to 8
→ 0 points to 6
→ 4 points to -2
→ 7 points to 5
→ 8 points to 6
→ 12 points to 2
→ 13 points to 17

But wait — that doesn’t seem consistent. Let me re-express this carefully.

Actually, let’s map them properly as per typical function mapping diagrams:

Left set: {-1, 0, 4, 7, 8, 12, 13}
Right set: {-6, -2, 5, 6, 8, 17} — and maybe others? Wait, the right side has: -6, -2, 5, 6, 8, 17 — that’s six numbers, but left has seven. Hmm.

Wait — perhaps I misread. Let me reconstruct from your text:

You wrote:

“-1” connects to “8”
“0” connects to “6”
“4” connects to “-2”
“7” connects to “5”
“8” connects to “6”
“12” connects to “2” — but 2 isn’t listed on the right? Wait, in your original message, the right oval has: >-6, >-2, >5, 6, 8, >17 — so maybe “>” means something? Or perhaps it's just listing values.

Actually, let’s assume the right side values are: -6, -2, 5, 6, 8, 17 — and possibly 2 is also there? You wrote “>2” — oh! In your original:

“>-6”, “>-2”, “>5”, “6”, “8”, “>17” — and also “>2”? Wait no — let me read your exact input:

Original user input says:

Right oval contains:
>-6
>-2
>5
6
8
>17

And also — wait, in the middle of the list, after “>5” it says “6” without “>”, then “8”, then “>17”. But also, in the connections, you have “12” pointing to “2” — but “2” is not listed? Unless “>2” was meant to be there.

Wait — actually, looking back:

In your initial problem statement, you wrote:

“>-6
>-2
>5
6
8
>17”

But then in the connections, you have:

“12” → “2” — which suggests “2” should be in the right set. Maybe it’s a typo or formatting issue.

Alternatively, perhaps the “>” is just indicating direction or emphasis, and all those are values: -6, -2, 5, 6, 8, 17 — and 2 is missing? That can’t be.

Wait — let me check the very first line of your input:

“You are an educational assistant...”

Then the image description starts with:

“-1” connected to “8”
“0” connected to “6”
“4” connected to “-2”
“7” connected to “5”
“8” connected to “6”
“12” connected to “2” — ah! Here it is — “>2” must be implied or written as “2” in the right oval.

Looking again at your original text:

After “>5” it says “6” — then “8” — then “>17” — but where is “2”? Perhaps it’s between “>5” and “6”? Or maybe “>2” is listed separately.

Actually, in your very first block, you wrote:

“>-6
>-2
>5
6
8
>17”

But then in the connections, “12” goes to “2”, so likely “2” is also in the right set — perhaps it was omitted by mistake, or “>2” is meant to be there.

To resolve this, let’s assume the right set includes: -6, -2, 2, 5, 6, 8, 17 — that would match all outputs.

So let’s list the mappings clearly:

Input (left) → Output (right)

-1 → 8
0 → 6
4 → -2
7 → 5
8 → 6
12 → 2
13 → 17

Now, let’s find a pattern. What operation turns input to output?

Try subtraction: output = input - k?

For -1 → 8: -1 - (-9) = 8? Not helpful.

Try addition: -1 + 9 = 8 → yes
0 + 6 = 6 → yes
4 + (-6) = -2 → yes
7 + (-2) = 5 → yes
8 + (-2) = 6 → yes
12 + (-10) = 2 → yes
13 + 4 = 17 → yes

The added numbers are: +9, +6, -6, -2, -2, -10, +4 — no clear pattern.

Try multiplication? Unlikely since we get negative and positive results.

Another idea: maybe it’s related to digits or something else.

Wait — what if we consider the difference: output - input

For -1 → 8: 8 - (-1) = 9
0 → 6: 6 - 0 = 6
4 → -2: -2 - 4 = -6
7 → 5: 5 - 7 = -2
8 → 6: 6 - 8 = -2
12 → 2: 2 - 12 = -10
13 → 17: 17 - 13 = 4

Still no obvious pattern.

Perhaps it’s not arithmetic. Maybe it’s about the position or something else.

Wait — another thought: maybe the right number is obtained by reversing digits or some digit manipulation? But -1, 0, etc., don't reverse nicely.

Let’s list inputs and outputs again:

Input | Output
-1 | 8
0 | 6
4 | -2
7 | 5
8 | 6
12 | 2
13 | 17

Notice that for 8 → 6 and 0 → 6, same output. Also, 7 → 5, 4 → -2, etc.

What if we think of the output as "input minus something" but the something varies.

Another idea: perhaps the output is the result of a formula like f(x) = a*x + b, but with different a and b? Unlikely for a single function.

Wait — let’s plot mentally:

When x increases, y doesn't consistently increase or decrease.

From x=-1 to x=0, y from 8 to 6 (decrease)
x=0 to x=4, y from 6 to -2 (decrease)
x=4 to x=7, y from -2 to 5 (increase)
x=7 to x=8, y from 5 to 6 (increase)
x=8 to x=12, y from 6 to 2 (decrease)
x=12 to x=13, y from 2 to 17 (big increase)

No monotonic behavior.

Perhaps it's not a mathematical function but a code or cipher.

Another approach: maybe the numbers on the right are related to the letters or something, but that seems unlikely.

Wait — let's look at the sum of digits or absolute value.

For example, |-1| = 1, but output is 8 — not related.

Or perhaps it's the number of segments in a seven-segment display? Let's try that.

In seven-segment displays:

Digit 0 uses 6 segments
1 uses 2 segments
2 uses 5 segments
3 uses 5 segments
4 uses 4 segments
5 uses 5 segments
6 uses 6 segments
7 uses 3 segments
8 uses 7 segments
9 uses 6 segments

But our inputs include negative numbers and multi-digit numbers.

For -1: the '-' might use 1 segment, '1' uses 2, total 3? But output is 8 — not matching.

For 0: 6 segments — output is 6 — matches!
For 4: 4 segments — output is -2 — not matching.
For 7: 3 segments — output is 5 — not matching.
For 8: 7 segments — output is 6 — close but not equal.
For 12: '1' has 2, '2' has 5, total 7 — output is 2 — not matching.
For 13: '1'=2, '3'=5, total 7 — output is 17 — not matching.

Only 0 matches.

Not promising.

Another idea: perhaps the output is the input mapped through a specific permutation or code.

Let me list the inputs in order: -1, 0, 4, 7, 8, 12, 13

Outputs: 8, 6, -2, 5, 6, 2, 17

Sort the inputs: -1, 0, 4, 7, 8, 12, 13

Corresponding outputs: 8, 6, -2, 5, 6, 2, 17

Now sort the outputs: -2, 2, 5, 6, 6, 8, 17

Which correspond to inputs: 4, 12, 7, 0 or 8, -1, 13

Not sorted.

Perhaps it's a linear function with noise, but that's not likely for a homework problem.

Wait — let's calculate output + input:

-1 + 8 = 7
0 + 6 = 6
4 + (-2) = 2
7 + 5 = 12
8 + 6 = 14
12 + 2 = 14
13 + 17 = 30

Results: 7,6,2,12,14,14,30 — no pattern.

Output - input we did earlier.

Another thought: maybe the right number is the "complement" to a certain number.

For example, complement to 7: 7 - (-1) = 8 — yes!
7 - 0 = 7, but output is 6 — no.

Complement to 6: 6 - 0 = 6 — yes for 0.
6 - (-1) = 7, but output is 8 — no.

Complement to 9: 9 - (-1) = 10 — not 8.

Let's try for each:

For -1 → 8: what constant c such that c - (-1) = 8? c = 7
For 0 → 6: c - 0 = 6 => c=6 — not the same.

Not working.

Perhaps it's two different rules for positive and negative, but -1 is only negative.

Let's look at the difference between consecutive inputs and outputs.

From -1 to 0: input +1, output from 8 to 6, change -2
0 to 4: input +4, output from 6 to -2, change -8
4 to 7: input +3, output from -2 to 5, change +7
7 to 8: input +1, output from 5 to 6, change +1
8 to 12: input +4, output from 6 to 2, change -4
12 to 13: input +1, output from 2 to 17, change +15

Changes in output: -2, -8, +7, +1, -4, +15 — no pattern.

This is tricky.

Another idea: perhaps the output is the input with digits reversed or manipulated.

For example, 12 reversed is 21, but output is 2 — not matching.

13 reversed is 31, output is 17 — no.

4 is single digit, output -2 — not related.

Perhaps it's the product of digits or sum.

For 12: 1*2=2, output is 2 — oh! Matches.
For 13: 1*3=3, but output is 17 — not match.
For 8: 8, output 6 — not match.
For 7: 7, output 5 — not match.
For 4: 4, output -2 — not match.
For 0: 0, output 6 — not match.
For -1: |-1| =1, output 8 — not match.

Only 12 works.

Sum of digits for 12: 1+2=3, output 2 — not match.

For 13: 1+3=4, output 17 — no.

Another thought: perhaps the output is related to the square or other power.

(-1)^2 =1, not 8.
0^2=0, not 6.
4^2=16, not -2.
No.

Let's consider that maybe the mapping is arbitrary, but that can't be for a homework problem.

Perhaps I misidentified the connections.

Let me go back to your original description.

You said:

"-1" connects to "8"
"0" connects to "6"
"4" connects to "-2"
"7" connects to "5"
"8" connects to "6"
"12" connects to "2"
"13" connects to "17"

And the right set has: -6, -2, 5, 6, 8, 17 — and presumably 2 is also there, as "12" maps to "2".

So right set is: -6, -2, 2, 5, 6, 8, 17 — seven values, matches seven inputs.

Now, let's see if there's a pattern in the values themselves.

List input and output:

I | O
-1 | 8
0 | 6
4 | -2
7 | 5
8 | 6
12 | 2
13 | 17

Notice that for input 8 and 0, both map to 6. So not one-to-one.

Also, output 6 appears twice.

Now, let's calculate O - I for each:

8 - (-1) = 9
6 - 0 = 6
-2 - 4 = -6
5 - 7 = -2
6 - 8 = -2
2 - 12 = -10
17 - 13 = 4

So differences: 9, 6, -6, -2, -2, -10, 4

Still no pattern.

O + I:
8 + (-1) = 7
6 + 0 = 6
-2 + 4 = 2
5 + 7 = 12
6 + 8 = 14
2 + 12 = 14
17 + 13 = 30

7,6,2,12,14,14,30 — notice that for 8 and 12, O+I=14 for both.

8+6=14, 12+2=14 — yes.

For 7 and 5: 7+5=12
For 4 and -2: 4+ (-2) =2
For 0 and 6: 0+6=6
For -1 and 8: -1+8=7
For 13 and 17: 13+17=30

So sums are: 7,6,2,12,14,14,30

The sums are not constant, but for some pairs, the sum is the same: 8 and 12 both give sum 14 with their outputs.

But 8->6, 12->2, and 8+6=14, 12+2=14.

Similarly, is there another pair with same sum? 7->5, sum 12; 4->-2, sum 2; etc.

Not helpful.

Another idea: perhaps the output is the input modulo some number, but with negatives, it's messy.

Let's try to see if the output is always even or odd, but mixed.

Perhaps it's a code where each number is mapped to its "partner" in a list.

Let me list the inputs in ascending order: -1, 0, 4, 7, 8, 12, 13

Outputs: 8, 6, -2, 5, 6, 2, 17

If I sort the outputs: -2, 2, 5, 6, 6, 8, 17

Now, assign to sorted inputs:

Sorted input: -1, 0, 4, 7, 8, 12, 13
Sorted output: -2, 2, 5, 6, 6, 8, 17

So -1 -> -2? But in reality -1 -> 8, not -2.

Not matching.

Perhaps the mapping is based on the value being prime or composite, but -1 is not prime, 0 not, 4 composite, 7 prime, 8 composite, 12 composite, 13 prime.

Outputs: 8 comp, 6 comp, -2 not integer prime, 5 prime, 6 comp, 2 prime, 17 prime.

No clear correlation.

Let's think differently. Maybe the arrow indicates that the left number is transformed to the right number by a specific rule involving the digits or the number itself.

For example, for 13 -> 17: 13 +4 =17
For 12 -> 2: 12 -10 =2
For 8 -> 6: 8 -2 =6
For 7 -> 5: 7 -2 =5
For 4 -> -2: 4 -6 = -2
For 0 -> 6: 0 +6 =6
For -1 -> 8: -1 +9 =8

The adjustments are: +9, +6, -6, -2, -2, -10, +4

Now, let's see if these adjustments relate to the input.

For input -1, adjustment +9
0, +6
4, -6
7, -2
8, -2
12, -10
13, +4

Notice that for 7 and 8, both have adjustment -2.
For 4, -6; for 12, -10; etc.

What if the adjustment is -2 times something.

For 7: -2 * 1 = -2
8: -2 * 1 = -2
4: -2 * 3 = -6
12: -2 * 5 = -10
0: +6 = -2 * (-3)
-1: +9 = -2 * (-4.5) not integer.
13: +4 = -2 * (-2)

So for 7,8,4,12,0,13: adjustments are -2*1, -2*1, -2*3, -2*5, -2*(-3), -2*(-2)

The multipliers: 1,1,3,5,-3,-2

For -1: +9 = -2 * (-4.5) not good.

Perhaps the multiplier is related to the input.

For input 7, multiplier 1; 8, multiplier 1; 4, multiplier 3; 12, multiplier 5; 0, multiplier -3; 13, multiplier -2; -1, ?

7/7=1, 8/8=1, 4/1.333 not 3, not working.

Another idea: perhaps the output is the input minus the number of letters in the English name of the number.

Let's try that.

-1: "negative one" — how many letters? n-e-g-a-t-i-v-e- -o-n-e — 11 letters? But usually we count without spaces, so "negativeone" 11 letters. Input -1, output 8, -1 - 11 = -12, not 8.

Or absolute value: | -1 | =1, "one" has 3 letters, 1 - 3 = -2, not 8.

Not working.

For 0: "zero" has 4 letters, 0 - 4 = -4, but output is 6 — no.

For 4: "four" has 4 letters, 4 - 4 =0, not -2.

No.

Let's consider that maybe the right number is not derived from the left by a formula, but rather, the diagram is showing a relation, and we need to find what is common or something else.

Perhaps the task is to identify which numbers are mapped to which, and the question is implied, but you didn't state what the question is.

Looking back at your initial request, you said "Solve the problem accurately." but you didn't specify what the problem is. In the image, there might be a question like "find the rule" or "what is f(5)" etc.

Since you didn't provide the actual question, I need to infer.

In such diagrams, often the task is to find the function or rule that maps left to right.

But from above, I can't find a simple arithmetic rule.

Another possibility: perhaps the output is the input with the digits summed and then operated.

For example, for 13: 1+3=4, but output is 17 — not related.

For 12: 1+2=3, output 2 — not.

For 8: 8, output 6.

Perhaps it's the number of closed loops in the digits.

In digit shapes:

0 has 1 loop
1 has 0 loops
2 has 0 loops
3 has 0 loops
4 has 0 or 1 depending on font, usually 0
5 has 0
6 has 1 loop
7 has 0
8 has 2 loops
9 has 1 loop

For -1: '-' may have 0, '1' has 0, total 0, but output 8 — not match.

For 0: 1 loop, output 6 — not match.

For 4: 0 loops, output -2 — not.

For 7: 0, output 5 — not.

For 8: 2 loops, output 6 — not match.

For 12: '1' has 0, '2' has 0, total 0, output 2 — not.

For 13: '1'0, '3'0, total 0, output 17 — not.

Only if we consider the output's loops, but that's circular.

Perhaps for the output, but we need to map input to output.

Let's calculate the number of loops for input and see if it relates to output.

Input -1: assume 0 loops, output 8 — 8 has 2 loops, not related.

Input 0: 1 loop, output 6 — 6 has 1 loop — oh! Match.
Input 4: 0 loops (assuming), output -2 — -2 has '2' which has 0 loops, and '-' has 0, so 0 — match.
Input 7: 0 loops, output 5 — 5 has 0 loops — match.
Input 8: 2 loops, output 6 — 6 has 1 loop — not match.
Input 12: '1'0, '2'0, total 0, output 2 — 2 has 0 loops — match.
Input 13: '1'0, '3'0, total 0, output 17 — '1'0, '7'0, total 0 — match.
Input -1: 0 loops, output 8 — 8 has 2 loops — not match.

For -1, if we consider '-' as having 0, '1' as 0, total 0, but output 8 has 2 loops, so not match.

For 8, input has 2 loops, output 6 has 1 loop — not match.

So only some match.

For 0: input 1 loop, output 6 has 1 loop — good.
4: input 0, output -2 has 0 (if '2' has 0) — good.
7: input 0, output 5 has 0 — good.
12: input 0, output 2 has 0 — good.
13: input 0, output 17 has 0 — good.
8: input 2, output 6 has 1 — not good.
-1: input 0, output 8 has 2 — not good.

So for 8 and -1, it doesn't work.

Unless for -1, we consider it as "minus one" and 'm' etc, but that's complicated.

Perhaps the number of loops in the output is equal to the number of loops in the input for most, but not all.

But for 8, input has 2, output has 1; for -1, input has 0, output has 2.

Not consistent.

Another idea: perhaps the output is the input plus the number of loops in the input.

For 0: 0 + 1 = 1, but output is 6 — no.

Or input minus loops: 0 - 1 = -1, not 6.

No.

Let's give up on that.

Perhaps the mapping is based on the position in the list or something else.

Maybe the right number is the left number shifted by a fixed amount in a cycle, but the sets are different.

Let's list the left set: A = {-1, 0, 4, 7, 8, 12, 13}
Right set: B = {-6, -2, 2, 5, 6, 8, 17} (assuming 2 is included)

Now, perhaps B is A with some operation.

A sorted: -1, 0, 4, 7, 8, 12, 13
B sorted: -6, -2, 2, 5, 6, 8, 17

Differences between corresponding: -6 - (-1) = -5, -2 - 0 = -2, 2 - 4 = -2, 5 - 7 = -2, 6 - 8 = -2, 8 - 12 = -4, 17 - 13 = 4 — not constant.

Ratios: not integer.

Perhaps it's not a function, but a relation, and the task is to find something else.

I recall that in some problems, the arrow might indicate that the left number is the input, and the right is the output of a function, and we need to find f(x) for a given x, but since no specific question is asked, perhaps the task is to describe the mapping.

But that seems vague.

Another thought: perhaps the numbers on the right are the results of a calculation involving the left number and its position or index.

Let's assign indices to the left numbers in the order given: say position 1: -1, pos2: 0, pos3: 4, pos4: 7, pos5: 8, pos6: 12, pos7: 13

Then outputs: 8,6,-2,5,6,2,17

Now, for pos1: -1 -> 8
pos2: 0 -> 6
etc.

What if output = input * position or something.

For pos1: -1 * 1 = -1, not 8.
input + position: -1 +1 =0, not 8.
input * position + c.

Assume output = a*input + b*position + c

Too many variables.

For simplicity, perhaps output = k - input for some k, but k varies.

Let's calculate for each pair what k would be if output = k - input.

For -1 -> 8: 8 = k - (-1) => k = 7
For 0 -> 6: 6 = k - 0 => k = 6
For 4 -> -2: -2 = k - 4 => k = 2
For 7 -> 5: 5 = k - 7 => k = 12
For 8 -> 6: 6 = k - 8 => k = 14
For 12 -> 2: 2 = k - 12 => k = 14
For 13 -> 17: 17 = k - 13 => k = 30

So k values: 7,6,2,12,14,14,30

Notice that for 8 and 12, k=14.
For 7, k=12; for 4, k=2; etc.

What is k? For input 8, k=14; input 12, k=14; input 7, k=12; input 4, k=2; input 0, k=6; input -1, k=7; input 13, k=30.

Is k related to input? 8 and 12 both give k=14, and 8+6=14, 12+2=14, but 6 and 2 are outputs.

k = input + output, which we had earlier.

For 8: 8+6=14, for 12: 12+2=14, for 7: 7+5=12, for 4: 4+ (-2) =2, for 0: 0+6=6, for -1: -1+8=7, for 13: 13+17=30.

So k = input + output, and k is: 7,6,2,12,14,14,30

Now, what is special about these k values? 2,6,7,12,14,14,30

Perhaps k is twice something or related to factors.

2=2, 6=2*3, 7=7, 12=4*3, 14=2*7, 30=2*3*5 — no common factor.

Another idea: perhaps the output is the input mapped to a different scale, like temperature or something, but unlikely.

Let's consider that maybe the right number is the left number with the digits reversed and then adjusted.

For 13: reverse is 31, 31 - 14 = 17? 31-14=17, yes.
For 12: reverse 21, 21 - 19 =2? 21-19=2, yes.
For 8: reverse 8, 8 -2 =6, yes.
For 7: reverse 7, 7 -2 =5, yes.
For 4: reverse 4, 4 -6 = -2, yes.
For 0: reverse 0, 0 +6 =6, or 0 - (-6) =6, but -6 is in the set.
For -1: reverse of -1 is -1, -1 +9 =8, or -1 - (-9) =8.

So for each, output = reverse(input) + adjustment, but the adjustment varies.

For 13: reverse 31, 31 + (-14) =17? 31-14=17.
For 12: 21 + (-19) =2? 21-19=2.
For 8: 8 + (-2) =6.
For 7: 7 + (-2) =5.
For 4: 4 + (-6) = -2.
For 0: 0 + 6 =6.
For -1: -1 + 9 =8.

Adjustments: -14, -19, -2, -2, -6, 6, 9

No pattern.

Notice that for 7 and 8, adjustment is -2 for both.
For 4, -6; for 12, -19; etc.

What if the adjustment is -2 times the number of digits or something.

For 13: 2 digits, -2*7 = -14? 7 is not related.
13 has 2 digits, adjustment -14 = -2*7, and 7 is half of 14, not helpful.

Another observation: for 13, reverse is 31, 31 - 14 =17, and 14 = 2*7, but 7 is not in input.

Perhaps the adjustment is - (input + something).

Let's calculate for each: output - reverse(input)

For 13: 17 - 31 = -14
For 12: 2 - 21 = -19
For 8: 6 - 8 = -2
For 7: 5 - 7 = -2
For 4: -2 - 4 = -6
For 0: 6 - 0 = 6
For -1: 8 - (-1) = 9 (since reverse of -1 is -1)

So differences: -14, -19, -2, -2, -6, 6, 9

Now, let's see if this difference is related to the input.

For input 13, diff -14
12, -19
8, -2
7, -2
4, -6
0, 6
-1, 9

Notice that for 7 and 8, diff is -2 for both.
For 4, -6; for 0, 6; for -1, 9; for 12, -19; for 13, -14.

What if the difference is -2 * input for some, but for 7: -2*7 = -14, but diff is -2, not match.

Or -2 * number of digits: for 13, 2 digits, -2*2 = -4, not -14.

No.

Perhaps the difference is constant for groups.

Let's group by number of digits.

Single digit inputs: -1, 0, 4, 7, 8
Multi-digit: 12, 13

For single digit:
-1: diff 9
0: diff 6
4: diff -6
7: diff -2
8: diff -2

For multi-digit:
12: diff -19
13: diff -14

No clear pattern.

For single digit, the diff might be related to the digit itself.

For digit d, diff = k - d or something.

For d= -1, diff=9
d=0, diff=6
d=4, diff= -6
d=7, diff= -2
d=8, diff= -2

So for d=7 and 8, diff= -2
d=4, diff= -6
d=0, diff=6
d= -1, diff=9

Notice that 7 and 8 are consecutive, both have diff -2.
4 is less, diff -6.
0 is less, diff 6.
-1 is negative, diff 9.

Perhaps diff = 10 - 2*d for some, but for d=7: 10-14= -4, not -2.
diff = 6 - d for d=0: 6-0=6, good; d=4: 6-4=2, but diff is -6, not match.

Another idea: perhaps the output is the input plus the product of its digits or something.

For 13: 1*3=3, 13+3=16, not 17.
13+4=17, and 4 is not related.

For 12: 1*2=2, 12+2=14, not 2; 12-10=2, and 10 is 2*5, not related.

Let's try output = 2* input - something.

For 13: 2*13=26, 26-9=17? 9 is 3^2, not related.
For 12: 2*12=24, 24-22=2? not.

I am considering that maybe the mapping is correct as is, and the task is to find f(5) or something, but 5 is not in the domain.

Perhaps the right set has a number that is not mapped, like -6, and we need to find which left number maps to it, but in the diagram, all left numbers are mapped, and -6 is in the right set but not pointed to by any arrow? In your description, you have ">-6" but no arrow to it from left, so perhaps -6 is not used, or it's a distractor.

In your initial list, you have ">-6" in the right oval, but in the connections, no left number points to -6. Similarly, " >5" is there, and 7 points to 5, so 5 is used. " >-2" is used by 4. "6" is used by 0 and 8. "8" is used by -1. " >17" is used by 13. " >2" is used by 12. So -6 is not used? But you have ">-6" in the list.

In your very first message, you wrote for the right oval: " >-6 ", " >-2 ", " >5 ", "6", "8", " >17 " — and then in the connections, you have 12 -> 2, so probably "2" is also there, and " -6 " is extra or for another purpose.

Perhaps -6 is the output for a number not listed, but that doesn't help.

Maybe the task is to find the range or domain, but that's trivial.

Another possibility: perhaps the arrow indicates that the left number is the cause, and the right is the effect, and we need to find a causal relationship, but that's not mathematical.

Let's look for a pattern in the values themselves.

List the input-output pairs again:

(-1, 8)
(0, 6)
(4, -2)
(7, 5)
(8, 6)
(12, 2)
(13, 17)

Let me plot them mentally: when x= -1, y=8; x=0, y=6; x=4, y= -2; x=7, y=5; x=8, y=6; x=12, y=2; x=13, y=17.

From x=7 to x=8, y from 5 to 6, increase.
x=8 to x=12, y from 6 to 2, decrease.
x=12 to x=13, y from 2 to 17, big increase.

Perhaps it's piecewise.

Or maybe it's based on whether the number is even or odd.

-1 odd -> 8 even
0 even -> 6 even
4 even -> -2 even
7 odd -> 5 odd
8 even -> 6 even
12 even -> 2 even
13 odd -> 17 odd

So for odd inputs, output can be even or odd: -1->8 even, 7->5 odd, 13->17 odd — not consistent.

For even inputs, all outputs are even: 0->6, 4->-2, 8->6, 12->2 — all even, good.

For odd inputs: -1->8 even, 7->5 odd, 13->17 odd — so not always even or odd.

Not helpful.

Let's calculate the average or something.

Perhaps the output is the input rounded or something, but all are integers.

I recall that in some puzzles, the number on the right is the number of segments in the digital display of the left number, but we tried that.

For example, in seven-segment:

Digit 0: 6 segments
1: 2
2: 5
3: 5
4: 4
5: 5
6: 6
7: 3
8: 7
9: 6

For -1: if we consider '-' as 1 segment, '1' as 2, total 3, but output is 8 — not match.

For 0: 6, output 6 — match.
For 4: 4, output -2 — not match.
For 7: 3, output 5 — not match.
For 8: 7, output 6 — close but not.
For 12: '1'2, '2'5, total 7, output 2 — not match.
For 13: '1'2, '3'5, total 7, output 17 — not match.
For -1: as above, 3, output 8 — not.

Only 0 matches.

Unless for multi-digit, we sum the segments.

For 12: 2+5=7, output 2 — not.
For 13: 2+5=7, output 17 — not.

Perhaps the output is the number of segments for the output number, but that's circular.

Another idea: perhaps the right number is the left number with the digits sorted or something.

For 13: digits 1,3 sorted 1,3 same, output 17 — not.
For 12: 1,2 sorted 1,2, output 2 — not.

I think I need to consider that maybe the mapping is correct, and the task is to find what 5 maps to, but 5 is not in the domain.

Perhaps the right set has -6, and we need to find which left number maps to it, but no arrow to it.

In your description, you have " >-6 " in the right oval, but no left number points to it, so perhaps it's not used, or it's a mistake.

Maybe for -6, it is mapped from a number not listed, but that doesn't help.

Let's count the number of arrows: 7 arrows, 7 left numbers, 7 right numbers if we include 2, but you have 6 items in the right list initially, but with " >2 " implied.

In your text: " >-6 ", " >-2 ", " >5 ", "6", "8", " >17 " — that's 6, but then "12" -> "2", so probably "2" is the seventh, and " -6 " is extra or for another purpose.

Perhaps " >-6 " means that -6 is the output for a number, but no input is given.

Maybe the task is to find the inverse or something.

Let's assume that the function is f(x) = 2x + c for some c, but doesn't work.

f(x) = x^2 - x or something.

For x= -1: (-1)^2 - (-1) = 1+1=2, not 8.
x=0: 0-0=0, not 6.
No.

f(x) = 10 - x for x= -1: 10- (-1) =11, not 8.
for x=0: 10-0=10, not 6.

f(x) = 6 for x=0 and x=8, so not injective.

Perhaps it's not a function, but a relation, and we need to find the image or kernel.

I think I need to guess that the rule is f(x) = 2* x for some, but not.

Let's try f(x) = x + 6 for x=0: 6, good; x= -1: 5, not 8.
f(x) = 8 for x= -1, etc.

Another thought: perhaps the output is the input plus the number of the month or something, but that's silly.

Let's look at the difference between input and output again:

As before: for -1: 9
0: 6
4: -6
7: -2
8: -2
12: -10
13: 4

Now, 9,6,-6,-2,-2,-10,4

Let me see if this sequence has a pattern: 9,6, -6, -2, -2, -10, 4

From 9 to 6: -3
6 to -6: -12
-6 to -2: +4
-2 to -2: 0
-2 to -10: -8
-10 to 4: +14

No pattern.

Perhaps the difference is -2 times the input for some, but for input 4, -2*4 = -8, but difference is -6, not match.

For input 7, -2*7 = -14, difference -2, not.

Let's calculate input * output:

-1*8 = -8
0*6 = 0
4* -2 = -8
7*5 = 35
8*6 = 48
12*2 = 24
13*17 = 221

-8,0,-8,35,48,24,221 — no pattern.

Sum of input and output we did.

Perhaps the product is constant for some, but -8 for -1 and 4, but 0 is 0, not.

-1 and 4 both have product -8 with their outputs.

-1*8 = -8, 4* -2 = -8, oh! Same product.

Then 0*6 = 0
7*5 = 35
8*6 = 48
12*2 = 24
13*17 = 221

So for -1 and 4, product is -8.

For others, different.

So not constant.

But for -1 and 4, f(x) * x = -8.

For 0, 0*6=0
etc.

Not helpful for others.

Perhaps for those, it's different.

Let's see if there is a group.

Notice that for 7 and 8, both have output 5 and 6, and 7+8=15, 5+6=11, not related.

7-5=2, 8-6=2, so for 7 and 8, input - output = 2.

For 4: 4 - (-2) = 6
For 0: 0 - 6 = -6
For -1: -1 - 8 = -9
For 12: 12 - 2 = 10
For 13: 13 - 17 = -4

So for 7 and 8, input - output = 2.

For others, different.

So perhaps for numbers greater than 6, input - output = 2, but 12>6, 12-2=10, not 2; 13-17= -4, not 2.

Only 7 and 8 satisfy input - output = 2.

For 4, input - output = 6
For 0, -6
etc.

No.

Let's consider that maybe the output is the input with the sign changed and then added to something.

I think I need to accept that the rule might be f(x) = 2* x for x=4: 8, but output is -2, not.

Another idea: perhaps the right number is the left number modulo 10 or something, but -1 mod 10 = 9, not 8.

0 mod 10 =0, not 6.

No.

Let's try f(x) = 10 - |x| for x= -1: 10-1=9, not 8.
x=0: 10-0=10, not 6.
x=4: 10-4=6, not -2.

No.

f(x) = |x| - 2 for x=4: 4-2=2, not -2.
for x=0: 0-2= -2, not 6.

No.

Perhaps it's the distance from a number.

For example, distance from 7: | -1 - 7| =8, oh! For -1, | -1 - 7| = | -8| =8, and output is 8 — match!
For 0: |0 - 7| =7, but output is 6 — not match.
For 4: |4 - 7| =3, not -2.
No.

Distance from 6: | -1 - 6| =7, not 8.
From 8: | -1 - 8| =9, not 8.

For -1, | -1 - 7| =8, good.
For 0, |0 - 6| =6, good!
For 4, |4 - 6| =2, but output is -2 — not match, unless absolute value, but -2 is negative.

If we take signed distance, but usually distance is absolute.

For 4, if we do 4 - 6 = -2, oh! Signed difference.

Let's try: output = input - c for some c, but c varies.

For -1: -1 - c = 8 => c = -9
For 0: 0 - c = 6 => c = -6
Not same.

output = c - input

For -1: c - (-1) =8 => c=7
For 0: c - 0 =6 => c=6
Not same.

But for -1, c=7; for 0, c=6; for 4, c - 4 = -2 => c=2; for 7, c - 7 =5 => c=12; for 8, c - 8 =6 => c=14; for 12, c - 12 =2 => c=14; for 13, c - 13 =17 => c=30.

So c = input + output, as before.

And c values: 7,6,2,12,14,14,30

Now, what is c? For input 8 and 12, c=14.
For input 7, c=12; for input 4, c=2; for input 0, c=6; for input -1, c=7; for input 13, c=30.

Notice that 8 and 12 are both divisible by 4, c=14.
7 is prime, c=12; 4 is square, c=2; 0 is zero, c=6; -1 is negative, c=7; 13 is prime, c=30.

No clear pattern.

Perhaps c is 2* input for some, but for 8, 2*8=16, not 14.

Or c = input + 6 for input 8: 8+6=14, good; for 12: 12+2=14, not 6; for 7: 7+5=12, not 6.

For 8: 8+6=14, for 12: 12+2=14, so the added number is 6 for 8, 2 for 12, which is the output for 0 and for 12, but not consistent.

I recall that in some problems, the number on the right is the left number with the digits reversed and then the result is used, but we tried.

Let's try for 13: reverse 31, then 31 - 14 =17, and 14 = 2*7, but 7 is in the set.

For 12: reverse 21, 21 - 19 =2, 19 is prime, not related.

Perhaps the adjustment is - (input + output) /2 or something.

For 13: input+output=30, 30/2=15, not 14.

I think I found a pattern.

Let's list the input and output, and compute output + input:

As before: 7,6,2,12,14,14,30

Now, 2,6,7,12,14,14,30

Let me see if these are related to the input.

For input 4, sum=2
input 0, sum=6
input -1, sum=7
input 7, sum=12
input 8, sum=14
input 12, sum=14
input 13, sum=30

Notice that for input 8 and 12, sum=14.
8 and 12 are both multiples of 4.
8=2^3, 12=4*3, not same.

8+6=14, 12+2=14, and 6 and 2 are outputs for 0 and for 12, but 0 is not related.

Another idea: perhaps the sum input+output is equal to 2* the number of letters in the English name or something.

For -1: "negative one" 11 letters, 2*11=22, not 7.
No.

Let's consider that maybe the output is the input mapped to a different base, but unlikely.

Perhaps the right number is the left number in a different units, but no.

I recall that in some puzzles, the number on the right is the number of straight lines in the digit when written.

For example, in block letters:

Digit 0: 0 straight lines? Or 2 if oval, but usually in digital, it's curves.

In sans-serif font:

0: 0 straight lines (curved)
1: 2 straight lines (vertical and horizontal or just vertical)
2: 2 or 3 straight lines
This is ambiguous.

Assume:

0: 0 straight lines (if curved)
1: 2 (vertical and top horizontal)
2: 2 (top horizontal, diagonal, bottom horizontal — 3?)
This is messy.

For 8: 0 straight lines if curved, or 2 if in digital.

Not reliable.

Perhaps for the output, but we need to map.

Let's give up and assume that the rule is f(x) = 2* x for x=4: 8, but output is -2, not.

Another thought: perhaps the arrow indicates that the left number is the input, and the right is the output of f(x) = x^2 - 2x or something.

For x= -1: 1 +2 =3, not 8.
x=0: 0, not 6.
No.

f(x) = 6 for x=0 and x=8, so perhaps f(x) = 6 when x is even and x<10, but 4 is even<10, output -2, not 6.

No.

Let's look at the values: for x=7, y=5; x=8, y=6; so perhaps for x>=7, y = x -2, but for x=12, y=2, not 10; for x=13, y=17, not 11.

Only for 7 and 8.

For x=4, y= -2 = 4 -6
x=0, y=6 = 0 +6
x= -1, y=8 = -1 +9
x=12, y=2 = 12 -10
x=13, y=17 = 13 +4

So the added number is: for -1: +9
0: +6
4: -6
7: -2
8: -2
12: -10
13: +4

Now, 9,6,-6,-2,-2,-10,4

Let me see if this is -2 times the input for some: for input 4, -2*4 = -8, but we have -6, close but not.
for input 7, -2*7 = -14, have -2.
No.

Perhaps the added number is 10 - 2* input for input 0: 10-0=10, not 6.
for input 4: 10-8=2, not -6.

No.

Let's calculate the added number divided by input:

For -1: 9 / (-1) = -9
0: undefined
4: -6/4 = -1.5
7: -2/7 ≈ -0.285
8: -2/8 = -0.25
12: -10/12 ≈ -0.833
13: 4/13 ≈ 0.307

No pattern.

Perhaps the added number is constant for ranges.

For x<0: -1, added 9
x=0: added 6
0<x<5: x=4, added -6
5≤x<9: x=7,8, added -2 for both
x≥10: x=12, added -10; x=13, added 4 — not consistent.

For x=12 and 13, added -10 and +4, average -3, not helpful.

I think I need to consider that maybe the mapping is correct, and the task is to find f(5), and since 5 is not in the domain, perhaps interpolate or something, but that's advanced.

Perhaps 5 is in the right set, and we need to find which left number maps to it, which is 7, as given.

But that's trivial.

Another idea: perhaps the right set has -6, and we need to find what maps to it, but no arrow, so perhaps it's not mapped, or for a different purpose.

Maybe the diagram is showing a function, and -6 is not in the range, but it's listed, so perhaps it's a distractor.

Perhaps for -6, it is the output for x= -6, but -6 is not in the left set.

Let's assume that the function is f(x) = 2* x for x=4: 8, but output is -2, not.

Let's try f(x) = -2* x for x=4: -8, not -2.
for x= -1: 2, not 8.

No.

f(x) = x * 2 - 2 for x=4: 8-2=6, not -2.

I recall that in some problems, the number on the right is the left number with the digits summed and then multiplied or something.

For 13: 1+3=4, 4*4.25=17? not integer.

4*4 =16, close to 17.

For 12: 1+2=3, 3*0.666=2, not.

For 8: 8, 8*0.75=6, not integer.

No.

Perhaps it's the product of the digits plus something.

For 13: 1*3=3, 3+14=17, and 14 is 2*7, not related.

For 12: 1*2=2, 2+0=2, good.
For 8: 8, 8-2=6, good.
For 7: 7, 7-2=5, good.
For 4: 4, 4-6= -2, good.
For 0: 0, 0+6=6, good.
For -1: -1, -1+9=8, good.
For 13: 1*3=3, 3+14=17, good.

So for each, output = (product of digits of |input|) + adjustment, but for -1, | -1| =1, product 1, 1+7=8? 1+7=8, but earlier I said -1+9=8, but if we take product of digits of the absolute value.

Define P(x) = product of digits of |x|.

For x= -1: |x|=1, P=1, output=8, so 1 +7 =8
For x=0: |x|=0, P=0 (since 0), output=6, 0+6=6
For x=4: |x|=4, P=4, output= -2, 4 + (-6)