Functions as Rules That Map Inputs to Outputs
Lumi stands at a large vending machine covered in transparent panels, tracing a finger along arrows that connect numbered buttons to snack slots, showing how each button press leads to exactly one item dropping out.
- Explain what makes a rule a function by describing the one-output-per-input condition in your own words.
- Identify whether a table, graph, or formula represents a function or a non-function and justify your answer.
- Compare a function to a non-function using a concrete mapping diagram, highlighting where the definition breaks down.
- Predict the output of a function given a specific input using a table, graph, or formula.
- Interpret function notation f(x) by correctly reading and evaluating expressions like f(3) or f(-1).
Key terms
- Function
- A rule assigning exactly one output to each input value.
- Domain
- The complete set of valid input values a function may receive.
- Range
- The complete set of output values a function actually produces.
- Vertical line test
- A graph is a function when no vertical line meets it more than once.
- Function notation
- The symbol f(x) naming the output produced from the input x.
The Uniqueness Condition
The defining property of a function is one-directional: each input must produce a single, unambiguous output, but outputs may be shared freely among inputs. A relation breaks the rule only when one input is paired with two or more different outputs. This asymmetry matters because functions are meant to be deterministic recipes — given the input, you can always predict the result without ambiguity. Constant functions, which send every input to the same output, are perfectly valid functions because no input is ever assigned two values.
Three Equivalent Representations
A function can be expressed as a table, a graph, or a formula, and each form encodes the same input-output rule. In a table, the test is whether any input value appears in two rows with different outputs. On a graph, the vertical line test captures the same idea visually, since a single vertical line represents one x-value and may legitimately touch the curve only once. In a formula such as f(x) = 2x + 1, substituting an input always returns one computed value, which is why algebraic formulas usually define functions automatically.
Evaluating With Function Notation
Function notation packages an instruction: f(3) means substitute 3 for every x in the rule, then simplify. Careful evaluation respects the order of operations and the sign of the input, so f(−1) for f(x) = 3x − 4 becomes 3(−1) − 4 = −7, not 3 − 4. Reading from a table works the same way: h(4) asks for the output recorded in the row where the input is 4. Treating the parentheses as multiplication, or swapping input and output columns, are the two most common evaluation slips.
Worked examples
Determine whether the relation {(1, 5), (2, 8), (1, 9)} is a function.
- List the inputs and check for repeats: the inputs are 1, 2, and 1, so 1 appears twice.
- Compare the outputs for the repeated input 1: they are 5 and 9, which differ.
- Because input 1 maps to two different outputs, the uniqueness condition fails.
Answer: It is not a function, since input 1 is paired with both 5 and 9.
Given f(x) = x² − 2x, evaluate f(-3).
- Substitute −3 for each x: f(−3) = (−3)² − 2(−3).
- Compute the square: (−3)² = 9.
- Compute the product: −2(−3) = +6, then add: 9 + 6 = 15.
Answer: f(-3) = 15.
Activity
Sort each relation card into the correct category — Function or Not a Function — by examining its table, graph, mapping diagram, or evaluated expression.
Practice
Decide whether the table x: 2, 4, 4, 6 with y: 1, 3, 7, 9 represents a function and justify your answer.
For g(x) = 5 − 2x, compute g(0), g(3), and g(-2), showing each substitution step.
Common mistakes to avoid
- Functions need distinct outputsDifferent inputs may share the same output; the rule only forbids one input from producing two different outputs.
- Parentheses in f(3) mean multiplyThe notation f(3) means evaluate the function at input 3, not multiply f by 3, so you substitute rather than multiply.
Check your understanding
Which of the following tables represents a function?
A student says: 'f(x) = x² cannot be a function because f(3) = 9 and f(−3) = 9, so two inputs give the same output.' What is wrong with this reasoning?
A relation is described by this information: a vertical line drawn at x = 2 intersects the graph at the points (2, 3) and (2, −3). What does this tell you about the relation?
Given f(x) = 3x − 4, what is the value of f(-1)?
The table below shows a function h. What is h(4)? x: 1, 2, 3, 4, 5 h(x): 10, 7, 4, 1, -2
Recap
A function assigns exactly one output to every input, a property checked by repeated-input tables or the vertical line test; the same rule appears as table, graph, or formula, and function notation like f(3) directs you to substitute the input and simplify to a single value.
Reflect
Which machines or processes in daily life behave like functions with predictable single outputs?