You can think of a function as a machine: you feed it one input value (\(x\)) and you get back exactly one output value (\(y\)). That is the key point: each input has exactly one output.

The precise definition: a function is a mapping from one set (the domain) to another set (the range) in which every element of the domain is paired with exactly one element of the range. Standard notation: \(y=f(x)\), where \(x\) is the independent variable and \(y\) is the dependent variable.

• Allowed: two different inputs leading to the same output (e.g., \(f(2)=5\) and \(f(3)=5\)).
• Not allowed: one input leading to two different outputs (e.g., \(f(2)=5\) and \(f(2)=7\)).

Vertical Line Test: a graph represents a function if and only if every vertical line (parallel to the \(y\)-axis) intersects it at most once. The idea is simple: a vertical line corresponds to a fixed \(x\) value, and if it meets the graph at two points it means the same \(x\) has two \(y\) values — contradicting the definition of a function.

Reading a graph: to find \(f(a)\), locate \(x=a\) on the horizontal axis, move vertically up to the graph, then move horizontally to the \(y\)-axis. The height you reach is the value of the function.

Note: a function does not have to be continuous. A graph with "jumps" can still be a function, as long as it passes the vertical line test.

Example 1: Identifying a Function from a Set of Ordered Pairs

Problem: Given the set \(\{(2,5),\,(3,5),\,(2,8)\}\). Does it represent a function?

Solution:

1. We check whether any input \(x\) repeats with different outputs.
2. The pair \((2,5)\) says \(f(2)=5\), but the pair \((2,8)\) says \(f(2)=8\).
3. The input \(x=2\) is paired with two different outputs, \(5\) and \(8\) — the uniqueness condition is violated.
4. The fact that \(f(2)=5\) and \(f(3)=5\) on its own is allowed (two inputs to the same output), but it does not save the situation.

Answer: No, the set does not represent a function, because the input \(x=2\) is paired with two different \(y\) values.

Example 2: Vertical Line Test on a Diagonal Line

Problem: Does the equation \(y=3x-2\) represent a function of \(x\)?

Solution:

1. The equation describes a diagonal line (slope \(3\), \(y\)-intercept \(-2\)).
2. For every \(x\) we substitute we get exactly one \(y\); for example \(x=1\) gives \(y=3\cdot 1-2=1\).
3. Vertical line test: every vertical line intersects a diagonal line at exactly one point.
4. Therefore the uniqueness condition holds.

Answer: Yes, \(y=3x-2\) represents a function of \(x\).

Example 3: Constant Function

Problem: Does \(y=4\) represent a function of \(x\)?

Solution:

1. The equation \(y=4\) describes a horizontal line at height \(4\).
2. For every \(x\) we choose, the output is always \(y=4\) — a single, well-defined output.
3. Vertical line test: every vertical line meets the horizontal line at exactly one point.
4. This is a constant function: \(f(x)=4\) for all \(x\).

Answer: Yes, \(y=4\) is a function (a constant function).

Example 4: Reading a Function Value from a Table

Problem: Given the table: \(x=-1\) is paired with \(y=4\); \(x=0\) is paired with \(y=1\); \(x=2\) is paired with \(y=5\). What is \(f(2)\)?

Solution:

1. We are looking for the output value corresponding to the input \(x=2\).
2. We locate in the table the row where \(x=2\).
3. That row shows \(y=5\).
4. Therefore \(f(2)=5\).

Answer: \(f(2)=5\).

Example 5: Circle — When a Relation Is Not a Function

Problem: Does the equation \(x^2+y^2=25\) (a circle of radius \(5\)) represent \(y\) as a function of \(x\)?

Solution:

1. We isolate \(y\): \(y^2=25-x^2\), giving \(y=\pm\sqrt{25-x^2}\).
2. Take \(x=3\) for example: we get \(y=\sqrt{16}=4\) and also \(y=-4\).
3. One input \((x=3)\) is paired with two outputs — the uniqueness condition is violated.
4. The vertical line test confirms this: a vertical line through the middle of the circle intersects it at two points.

Answer: No, the circle does not represent \(y\) as a function of \(x\).

✗ Common mistake: Students think that if one output is produced by several different inputs, then it is not a function.

✓ The correct way: The restriction goes in one direction only: a single input must not produce several outputs. But it is perfectly fine for two different inputs to lead to the same output. For example \(f(x)=x^2\) satisfies \(f(2)=f(-2)=4\) — and it is a valid function.

✗ Common mistake: Students think a graph with a jump or a break cannot be a function.

✓ The correct way: Continuity is not a requirement for a function. A broken graph can still be a function as long as it passes the vertical line test — that is, every \(x\) yields at most one \(y\) value.

✗ Common mistake: When reading a value from a graph, students confuse the axes and read \(x\) instead of \(y\).

✓ The correct way: To find \(f(a)\): start at \(x=a\) on the horizontal axis, move vertically up to the graph, then move horizontally to the \(y\)-axis. The value you read on the \(y\)-axis is the output.

• Function: every input \(x\) is paired with exactly one output \(y=f(x)\).
• Vertical Line Test: a graph is a function if every vertical line intersects it at most once.
• Two different \(x\) values may share the same \(y\); one \(x\) may not have two \(y\) values.
• A function does not have to be continuous.
• Reading \(f(a)\): from \(x=a\) move up to the graph and then across to the \(y\)-axis.