Are these relations functions?

Let’s go step by step to figure out which diagrams show functions.

What is a function?

A function is a special kind of relationship between two sets (we call them input and output).
The rule is:
→ Each input can only have ONE output.
It’s okay if multiple inputs go to the same output — that’s fine!
But if one input goes to MORE than one output, then it’s NOT a function.

Think of it like this:
If you’re at a vending machine, and you press button “A”, you should always get the same snack. If pressing “A” sometimes gives you chips and sometimes gives you candy, that’s not reliable — just like not being a function!

---

Now let’s look at each diagram in order (left to right, top to bottom). We’ll check if any input has more than one arrow going out from it.

We’ll label rows 1 to 5 (top to bottom), and columns 1 to 4 (left to right).

---

Row 1:

- Diagram 1: Input -3 → outputs -2 AND -1 → ✘ Not a function
- Diagram 2: Each input (-3,-2,-1,0) has exactly one output → ✔ Function
- Diagram 3: Same as #2 → ✔ Function
- Diagram 4: Input -3 → outputs -2 AND -1 → ✘ Not a function

Row 2:

- Diagram 1: Input -1 → outputs 1 AND 2 → ✘ Not a function
- Diagram 2: All inputs have one output → ✔ Function
- Diagram 3: All inputs have one output → ✔ Function
- Diagram 4: All inputs have one output → ✔ Function

Row 3:

- Diagram 1: Input -2 → outputs 1 AND 2 → ✘ Not a function
- Diagram 2: All inputs have one output → ✔ Function
- Diagram 3: Input 0 → outputs 1 AND 2 → ✘ Not a function
- Diagram 4: Input -2 → outputs 1 AND 2 → ✘ Not a function

Row 4:

- Diagram 1: All inputs have one output → ✔ Function
- Diagram 2: All inputs have one output → ✔ Function
- Diagram 3: All inputs have one output → ✔ Function
- Diagram 4: Input -2 → outputs 1 AND 2 → ✘ Not a function

Row 5:

- Diagram 1: Input -1 → outputs 1 AND 2 → ✘ Not a function
- Diagram 2: All inputs have one output → ✔ Function
- Diagram 3: Input 0 → outputs 1 AND 2 → ✘ Not a function
- Diagram 4: All inputs have one output → ✔ Function

---

Now let’s count how many are functions:

✔ Row 1: #2, #3 → 2
✔ Row 2: #2, #3, #4 → 3
✔ Row 3: #2 → 1
✔ Row 4: #1, #2, #3 → 3
✔ Row 5: #2, #4 → 2

Total = 2 + 3 + 1 + 3 + 2 = 11

Wait — let me double-check by listing all the “Yes” ones:

From above:

Row 1: 2, 3
Row 2: 2, 3, 4
Row 3: 2
Row 4: 1, 2, 3
Row 5: 2, 4

That’s:
(2,3) → 2
(2,3,4) → 3
(2) → 1
(1,2,3) → 3
(2,4) → 2
Total: 2+3=5; 5+1=6; 6+3=9; 9+2=11 → Yes, 11 total.

But wait — the question says “Are these functions?” and there are 20 diagrams (5 rows × 4 columns). So we’re counting how many of the 20 are functions.

Final count: 11

---

Final Answer:
11