Key takeaway: A function is a dependable matching in which each input is paired with exactly one output, so no input ever leads to two different answers.
Grade 88.F.A.1Functions and Graphs
Definition of Function
A function is a dependable machine: feed in a value and it hands back one specific answer, every time. Allowed inputs make up the domain, and the answers that come back make up the range. Buying gas works this way, because any number of gallons has a single total price.
The one requirement concerns inputs. An input may not be tied to two different outputs. An output may be shared, however; two different inputs can lead to the same answer without breaking anything. The rule can be shown as a table, a list of ordered pairs, a mapping diagram, a graph or an equation, and each form lets you test it in its own way.
Formula and notation
Write the rule with a name, an input and an output:
\[y = f(x) \qquad \text{for example,} \qquad f(x) = 4x + 1\]
Here \(f(2) = 9\). A relation is a function exactly when no \(x\) value repeats with different \(y\) values.
Key facts
- Each input has one and only one output.
- Two inputs can share the same output without violating the rule.
- On a graph, a vertical line never touches the curve twice.
- Inputs form the domain and outputs form the range.
Function example
Two lists of (input, output) pairs come from a vending machine and a faulty one. Test both: \(\{(1, 6), (2, 9), (3, 6), (4, 12)\}\) and \(\{(5, 2), (5, 7), (8, 1)\}\).
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First set: no first coordinate is repeated, so each input is matched with one output.
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The output 6 appears twice, but it comes from two separate inputs, which is allowed.
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Second set: the first coordinate 5 shows up with 2 and with 7. One input leads to two outputs.
The first set is a function; the second set is not.
Common mistake
Watch out
Nia sees that 6 is repeated in the outputs of the first set and says it cannot be a function.
Why it is wrong
Repeated outputs are fine. A function only forbids a repeated input with different outputs. Here 1 and 3 both lead to 6, but each of them leads nowhere else. Scan the first coordinates for duplicates, not the second.
Related terms and lessons
- Grade 8 · Introduction to FunctionsLessonWorksheetQuiz
- Algebra 1 · Functions, Domain, Range and Function NotationLessonWorksheetQuiz
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