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Stage 4 · Easier Concepts You Might Not Know
Recognizing Functions
Each input can only have one output. Learn to spot it in tables and graphs — a key GED skill that tests logic and pattern recognition, not difficult calculations.
📖 What Is a Function?
A function is a rule where each input has exactly one output. That's the whole definition. You don't need to find a formula — you just need to check: does any one input have more than one output?
Input = x value. Output = y value.
The Rule
One x → one y = function ✅
One x → two different y values = NOT a function ❌
Outputs CAN Repeat
Repeated y values are fine. Only repeated x values with different outputs are a problem.
👤 Think of It Like Ages
Think of each person as an x value (input) and their age as a y value (output).
Many people can be the same age. Outputs can repeat — no problem. But one person cannot be two different ages at the same time. That would be lying about your age.
The key test: If one x value has two DIFFERENT y values, it's like one person having two ages at once. Impossible. That means it's NOT a function.
📋 Table Strategy — Look for Duplicate Inputs
When you see a table, scan the x column and follow these three steps:
The 3-Step Table Check
Step 1 — Scan only the x column for any duplicate values
Step 2 — No duplicates? → It IS a function. Done.
Step 3 — Duplicates found? → Check if their y values are the same or different
Same output = still a function ✅ Different output = NOT a function ❌
Ignore the y column. Repeated y values never cause a problem. Only the x column matters.
✅ IS a function — all x unique
x
y
1
5
2
5
3
5
4
8
5
8
6
8
Y values repeat — that's totally fine.
❌ NOT a function — x = 2 has two outputs
x
y
1
3
2
5
2
8
3
7
4
9
5
11
x = 2 gives 5 AND 8. One person, two ages.
✅ IS a function — duplicate x, same output
x
y
1
4
2
6
3
8
3
8
4
10
5
12
x = 3 repeats, but both give y = 8. Still fine.
📈 Graph Strategy — The Vertical Line Test
For graphs, use the Vertical Line Test. Imagine drawing vertical lines from top to bottom anywhere across the graph.
The Vertical Line Test
Every vertical line touches the graph exactly once → IS a function ✅
Any vertical line touches the graph more than once → NOT a function ❌
✅ Diagonal line — function
Each vertical line hits once.
❌ Circle — not a function
Vertical line crosses twice — fails!
✅ U-parabola — function
Each vertical line hits once. Curves can be functions!
❌ Sideways parabola — not a function
Vertical line crosses twice — fails!
⭕ Open and Closed Circles on Step Graphs
Some graphs use open circles (○) and closed circles (●) at endpoints. These matter when two segments meet at the same x value.
● Closed = included
The point IS part of the graph. That x value has that y value.
○ Open = not included
The point is NOT part of the graph. The segment stops just before that x value.
At a junction: Open circle ending + closed circle starting at the same x → only one segment owns that point → still a function. Two closed circles at the same x → two outputs → NOT a function.
✅ Step graph — IS a function
At x = 2: first segment ends with open circle (not included), second begins with closed circle. x = 2 has only one output.
❌ Step graph — NOT a function
At x = 2: both segments have closed circles. x = 2 has two outputs (y = 2 and y = 4) → NOT a function.
⚠️ Common Mistakes
Students often think these are problems — they're NOT:
• Repeated y values — outputs can repeat freely
• Curved graphs — curves can absolutely be functions
• Negative x or y values — totally fine in functions
• The graph going up AND down — doesn't affect the function rule
Remember: Many people can be the same age. But one person cannot be two different ages at the same time.
✏️ Practice Questions
Bank 1 — Guided Practice
Questions include hints built into the wording — great for learning the concept
You've answered 5 questions. Keep going or check your score.
Bank 2 — GED-Level Practice
Independent practice — apply the rules on your own
You've answered 5 questions. Keep going or check your score.