Identify functions

1. A relation is a function when no input is paired with two different outputs.
2. Check each set's inputs (first numbers):
   • \{(1, 4), (2, 5), (3, 6), (4, 7)\}: inputs 1, 2, 3, 4 — all different ✓
   • \{(1, 2), (1, 3), (2, 4), (3, 5)\}: input 1 → 2 and 3 ✗
   • \{(2, 1), (3, 1), (2, 5), (4, 1)\}: input 2 → 1 and 5 ✗
   • \{(0, 6), (5, 7), (0, 8), (6, 9)\}: input 0 → 6 and 8 ✗
3. Only the first set gives each input exactly one output.

Answer: \{(1, 4), (2, 5), (3, 6), (4, 7)\}

Check: For each input in that set you can name one output without hesitation: 1→4, 2→5, 3→6, 4→7 ✓

Tip: Scan the inputs first. If no input repeats, you are done — it is a function.

1. The inputs are 2, 4, 4, 6. The input 4 appears twice.
2. Its outputs are 7 (in the second column) and 9 (in the third column).
3. One input with two different outputs breaks the definition of a function.
4. The pattern in the y-row (going up by 2) does not matter.

Answer: No, because x = 4 is paired with two different y-values.

Check: Plot (4, 7) and (4, 9) — they lie on the same vertical line x = 4, so the vertical line test fails ✓

Tip: A nice pattern in the outputs is not evidence of a function. Only the one-output-per-input rule decides.

1. A relation is a function when each input (x, the first number) has exactly one output.
2. Check the inputs of each set:
   • \{(-2, 3), (-1, 3), (0, 3), (1, 3)\}: inputs -2, -1, 0, 1 — all different ✓ (the same output 3 is allowed).
   • \{(4, -1), (6, 0), (8, 1), (10, 2)\}: inputs 4, 6, 8, 10 — all different ✓.
   • \{(7, 2), (-3, 5), (7, -2), (1, 0)\}: input 7 appears with 2 and with -2 ✗.
   • \{(0, 0), (1, 1), (-1, 1), (2, 4)\}: inputs 0, 1, -1, 2 — all different ✓.
3. Only \{(7, 2), (-3, 5), (7, -2), (1, 0)\} has an input with two outputs.

Answer: \{(7, 2), (-3, 5), (7, -2), (1, 0)\}

Check: Ask "What is the output for 7?" — the set gives two answers, 2 and -2. A function can never do that ✓.

Tip: Repeated outputs are fine. Only a repeated input with different outputs breaks a function.

1. To decide, look at the inputs in the x row: -3, -1, 0, 1, 3.
2. Every input appears once, so each input has exactly one output.
3. The outputs 9 and 1 each appear twice, but that is allowed: different inputs may share an output. (-3 and 3 both give 9 — this table follows y = x^2.)
4. The order of the outputs does not matter either.

Answer: Yes, because each x-value is paired with exactly one y-value.

Check: Name the output for every input without hesitation: -3 \to 9, -1 \to 1, 0 \to 0, 1 \to 1, 3 \to 9 — one answer each ✓.

Tip: For a table, scan only the input row for repeats. If no input repeats, the table is a function — no matter what the outputs do.

1. Gallons → total cost at $3.60 per gallon: each number of gallons has one cost (C = 3.60g) → function.
2. Date → that day's high temperature in a city: each day has one recorded high → function.
3. Age in years → height of an adult: two 30-year-olds can have different heights, so the input 30 can have many outputs → not a function.
4. Side length of a square → its area: each side length s gives one area s^2 → function.

Answer: input = an adult's age in years, output = that person's height

Check: Find a counterexample: two people, both 30 years old, one 5 ft 4 in and one 6 ft 1 in — one input, two outputs ✓

Tip: In real-world questions, try to imagine two different outputs for the same input. If you can, it is not a function.

1. Write the points: (-4, 1), (-2, 3), (0, -2), (2, 3), (2, -1), (4, 0).
2. Inputs: -4, -2, 0, 2, 2, 4. The input 2 appears twice.
3. Its outputs are 3 and -1 — different. A vertical line at x = 2 passes through both points.
4. (The output 3 appears twice, at x = -2 and x = 2 — that alone would be fine.)

Answer: No, because the points (2, 3) and (2, -1) have the same x-value but different y-values.

Check: Vertical line test: the line x = 2 touches two points ✓ — fails.

Tip: For scatter plots, look for dots directly above one another. Dots side-by-side at the same height are fine.

1. List the points: (-3, 2), (-1, 4), (0, -1), (2, 3), (2, -2), (4, 1).
2. Look for a repeated x-value: x = 2 appears twice, with y = 3 and y = -2.
3. That pair makes the relation fail: a vertical line at x = 2 hits two points.
4. Removing either (2, 3) or (2, -2) fixes it. Among the choices, only (2, -2) is one of these points.

Answer: (2, -2)

Check: Without (2, -2) the x-values are -3, -1, 0, 2, 4 — all different, so every vertical line meets at most one point ✓.

Tip: For a graph of separate points, the vertical line test still works: two dots stacked in the same column mean "not a function."

1. Weight of a package → shipping cost: the table gives exactly one price for any weight (for example 12 oz → $6.75) → function.
2. A person's first name → their phone number: many people are named Maria, with different numbers → not a function.
3. A ZIP code → a person who lives in that ZIP code: thousands of people share one ZIP code → not a function.
4. A month → a day in that month when it rained: one month can have many rainy days → not a function.

Answer: input = a package's weight; output = its shipping cost from the table

Check: Different weights may share a price (10 oz and 15 oz both cost $6.75). That is allowed — two inputs, same output ✓.

Tip: In word problems, try to imagine one input with two outputs. If you can, it is not a function.

1. Count the arrows leaving each input.
2. Diagram A: input 1 has two arrows (to 4 and to 9) ✗.
3. Diagram B: 2 → 3, 4 → 5, 6 → 3, 8 → 5. Each input has exactly one arrow ✓. (Two arrows arriving at 3, and two at 5, is fine.)
4. Diagram C: input 5 has two arrows (to 2 and to 3) ✗.
5. Diagram D: input 10 has two arrows (to 20 and to 40) ✗.

Answer: Diagram B

Check: Write Diagram B as pairs: (2, 3), (4, 5), (6, 3), (8, 5). The inputs 2, 4, 6, 8 are all different ✓.

Tip: In a mapping diagram, look only at the left side: one arrow out of every input means a function. Many arrows into one output is allowed.

1. The inputs are 4, 5 and 6 — no input repeats.
2. Each input has exactly one output: 4 \to 1, 5 \to 1, 6 \to 1.
3. The rule for a function only limits inputs: an input may not have two outputs. Many inputs sharing the same output is allowed.
4. So the relation is a function, and Ben's reason is the correct one. Ava's reason confuses repeated outputs with repeated inputs.
5. These points lie on the horizontal line y = 1, which passes the vertical line test.

Answer: Ben

Check: A vertical line through x = 4, 5, or 6 meets only one point each time ✓.

Tip: "Every input gives the same output" describes a constant function — still a function.

1. x^2 + y^2 = 25 — at x = 0: y^2 = 25, so y = 5 or y = -5 ✗ (a circle).
2. y^2 = x + 4 — at x = 0: y^2 = 4, so y = 2 or y = -2 ✗ (a sideways parabola).
3. x = 7 — the single input 7 pairs with every y ✗ (a vertical line).
4. y = -2x^2 + 3 — for any x you square it, multiply by -2, and add 3: exactly one result ✓ (a parabola opening down).

Answer: y = -2x^2 + 3

Check: Try x = 1 and x = -1: both give y = -2 + 3 = 1. Two inputs sharing one output is allowed, and each input still has one output ✓.

Tip: If an equation is already solved as "y = (expression in x)", it is a function. If y is squared and cannot be isolated without \pm, it is not.

1. y = x^2 - 4 — one y for each x (a parabola opening up) → function.
2. y = 5 — every x gives y = 5 (horizontal line) → function.
3. x + y^2 = 9 — solve: y^2 = 9 - x, so y = \pm\sqrt{9 - x}. For x = 0: y = 3 or y = -3 → not a function.
4. y = \lvert x - 2 \rvert — one y for each x (a V-shape) → function.

Answer: x + y^2 = 9

Check: Both (0, 3) and (0, -3) satisfy x + y^2 = 9: 0 + 9 = 9 ✓ and 0 + (-3)^2 = 9 ✓ — one input, two outputs.

Tip: When y is squared (and you cannot get rid of it), solving for y gives \pm, which means two outputs — not a function.

1. Relation 1 (friend → month): Lena → March, Omar → July, Priya → March, Theo → October. Each friend has exactly one birth month → function. (Two friends share March, which is a repeated output — allowed.)
2. Relation 2 (month → friend): March → Lena and March → Priya. The input March has two outputs → not a function.
3. So Relation 1 is a function, but Relation 2 is not.

Answer: Relation 1 is a function, but Relation 2 is not.

Check: Ask Relation 2 "which friend was born in March?" — two possible answers ✓ Ask Relation 1 "when was Priya born?" — exactly one answer, March ✓

Tip: Reversing a function (swapping inputs and outputs) turns repeated outputs into repeated inputs — so the reverse may not be a function.

1. The definition: a function gives each input exactly one output. So the check is on the x row.
2. Marcus went wrong at Step 1: he looked at the y-values (outputs) instead of the x-values (inputs).
3. Correct check: inputs 1, 2, 3, 4, 5 — none repeats. So each input has one output.
4. The repeated outputs 7 and 8 are allowed. The table is a function, so Step 3's conclusion is also wrong — it follows from the wrong check.

Answer: Step 1 is the error: he should check the x-values. The table is a function.

Check: A vertical line at each x from 1 to 5 hits exactly one point of the table's graph ✓.

Tip: In "find the error" questions, find the first step that goes wrong. Every later step built on it is suspect too.

1. List the points: (62, 7), (64, 8), (65, 9), (68, 9), (68, 10.5), (71, 11), (74, 12).
2. The height 68 inches appears twice, with shoe sizes 9 and 10.5.
3. One input (68 in) with two different outputs means the data do not form a function.
4. The shoe size 9 appears for heights 65 and 68 — two inputs sharing an output is allowed and is not the reason.
5. The upward trend does not matter; the definition is about one output per input.

Answer: No, because the height 68 inches is paired with two different shoe sizes.

Check: A vertical line at x = 68 passes through two dots, (68, 9) and (68, 10.5) ✓.

Tip: Real data often fail to be functions because two people (or days, or items) share an input. Check for stacked dots.

1. Relation I (stop → time): the inputs are 1, 2, 3, 4, 3, 1. Stop 3 has two times (7:20 and 7:40), and stop 1 has two times (7:05 and 7:52) → not a function.
2. Relation II (time → stop): the inputs are the six times, which are all different. Each time has exactly one stop → function.
3. In Relation II, stops 1 and 3 appear twice as outputs, which is allowed.

Answer: Only Relation II is a function.

Check: Ask Relation I "When does the bus reach stop 3?" — two answers. Ask Relation II "Where is the bus at 7:40?" — one answer, stop 3 ✓.

Tip: Swapping input and output can change the answer. Repeated values in a column only matter when that column is the input.

1. x = total cost of an order, y = number of pizzas: a $30 order could be 3 small pizzas (3 \times 10) or 2 large pizzas (2 \times 15). One input, two outputs → not a function.
2. x = number of large pizzas only, y = total cost: y = 15x — each number of pizzas has exactly one cost → function.
3. x = pizza size, y = a topping chosen: a large pizza can have pepperoni or mushrooms → not a function.
4. x = delivery time in minutes, y = the address delivered to: two orders can both take 25 minutes to different addresses → not a function.

Answer: x = the number of large pizzas in an order of only large pizzas; y = the total cost

Check: Test inputs: 1 pizza → $15, 2 → $30, 3 → $45. No input ever gets two costs ✓.

Tip: A rule like y = 15x always makes a function. If the input could come from several different situations, look for a counterexample.

1. The first column is the pair (a, 10). The relation fails only if a equals another input whose output is not 10.
2. a = 3: pairs (3, 10) and (3, 4) — input 3 has two outputs → not a function ✓.
3. a = 6: pairs (6, 10) and (6, 10) — the same pair twice; input 6 still has only the output 10 → function.
4. a = 10: 10 is a new input, so (10, 10) causes no clash → function.
5. a = 0: 0 is a new input → function.

Answer: a = 3

Check: With a = 3, asking "what is y when x = 3?" gets two answers, 10 and 4 ✓. (a = 9 would also fail, but it is not a choice.)

Tip: A repeated input only breaks a function when its outputs differ. The trap here is a = 6, which repeats the input but keeps the same output.

1. "If a relation is a function, then no output repeats." False: \{(1, 5), (2, 5)\} is a function with a repeated output.
2. "If a relation has an input with two different outputs, then its graph fails the vertical line test." True: the two points (a, b) and (a, c) lie on the same vertical line x = a, so that line meets the graph twice.
3. "Every straight line in the coordinate plane is the graph of a function." False: a vertical line such as x = 2 is not a function.
4. "If no output repeats, then the relation is a function." False: \{(1, 2), (1, 3)\} has different outputs but the input 1 repeats.

Answer: If a relation has an input with two different outputs, then its graph fails the vertical line test.

Check: The vertical line test is just the picture version of the definition: same x (one vertical line), two y's (two crossing points) ✓.

Tip: For "always true" questions, one counterexample is enough to reject a statement.