Math 10C · Relations and functions
What makes a relation a function
A relation is any pairing of inputs with outputs. A function is a relation that never gives two answers to the same question — and that one restriction is what makes the whole of the rest of mathematics possible.
- 1. The words, first
- 2. The rule, and why it exists
- 3. Domain and range
- 4. Function notation
- 5. What costs marks
The words, first
The idea: Domain and range describe different axes, and function notation is an instruction rather than a multiplication.
| Word | What it means |
|---|---|
| Relation | Any set of ordered pairs — any pairing of inputs with outputs. |
| Function | A relation in which each input has exactly one output. Outputs may be shared; inputs may not. |
| Domain | The set of allowed inputs. The x-values. |
| Range | The set of outputs reached. The y-values. |
| Vertical line test | A graph is a function if no vertical line crosses it more than once, because a vertical line is a single input. |
| Function notation | f(x) = 3x − 4 names the rule. f(2) means substitute 2 for x, not f times 2. |
| Independent variable | The input, usually x, plotted horizontally. |
| Dependent variable | The output, usually y, plotted vertically — it depends on the input. |
| Discrete / continuous | Separate points, or an unbroken line. The situation decides which is appropriate. |
The rule, and why it exists
The idea: If one input could give two outputs, no prediction would ever be reliable. The function requirement is what buys you the right to say “so the answer is”.
The test. {(1, 2), (1, 3), (2, 4)} is not a function: the input 1 appears twice with different outputs. {(1, 5), (2, 5), (3, 5)} is a function — repeating an output is allowed, and every input still has exactly one answer.
Graphically that becomes the vertical line test. A circle fails it; a parabola opening upward passes it; a vertical line fails it as badly as possible.
Why it matters. Ask what a taxi ride costs and there had better be one answer. A relation that gave two would be useless for planning anything, which is why the objects worth studying are functions.
Domain and range
The idea: Read the graph left to right for domain and bottom to top for range. In a word problem, the situation restricts them further than the algebra does.
From a graph. A line with no restrictions has domain and range of all real numbers. A parabola with vertex (2, −3) opening upward has domain all real numbers and range y ≥ −3, because the vertex is the lowest point reached.
From a situation. The cost of n tickets at $12 each has a domain of whole numbers from 0 upward — you cannot buy half a ticket or a negative number of them. Saying that explicitly is usually worth a mark, and noticing the data is discrete rather than continuous is part of the same point.
Notation. Domain and range can be written as an inequality (x ≥ 0), in set notation ({x | x ≥ 0, x ∈ ℝ}) or in interval notation ([0, ∞)). Any is acceptable unless the question asks for one.
Function notation
The idea: f(x) is a machine with a name. f(2) says put 2 in; f(x + 1) says put the whole expression in.
Substituting a number. For f(x) = 3x − 4, f(2) = 3(2) − 4 = 2.
Substituting carefully. For f(x) = x² − 2x, f(−3) = (−3)² − 2(−3) = 9 + 6 = 15. Both minus signs are easy to lose; brackets prevent it.
Substituting an expression. For f(x) = 2x + 5, f(x + 1) = 2(x + 1) + 5 = 2x + 7. The 2 multiplies the whole input, not just the x.
Going backwards. If f(x) = x² + 1 and f(a) = 10, then a² = 9, so a = 3 or −3. Squaring loses the sign, so both survive.
Reading a graph in notation. f(3) = 5 means the point (3, 5) is on the graph. That translation is worth being fluent in, because exam questions switch between the two freely.
What costs marks
The idea: Two are notation and two are reading the wrong axis.
- Reading f(2) as f × 2.
- Swapping domain and range. Alphabetical works: d before r, x before y.
- Ignoring the situation when a word problem restricts the domain.
- Giving one answer when solving f(a) = k produces two.