Mathematics / Lesson 02
Functions as dependence
Read a function as a dependable relationship between changing quantities.
orient
Why this idea had to exist
A function is useful whenever one choice determines another value: time determines position, an image coordinate determines color, and a model input determines a prediction. The notation matters less than the dependency it records.
intuition
Build a picture you can reason with
Picture a well-built instrument with one input dial and one output needle. Each allowed dial position produces exactly one needle reading. The instrument may be smooth, jumpy, or impossible to reverse; it is still a function if the same input never gives two outputs at once.
formalize
Give the intuition a precise edge
A function f maps every element of a domain to exactly one element of a codomain. The value f(x) is the output assigned to x. Composition g(f(x)) connects instruments in sequence; an inverse reverses the mapping only when outputs identify inputs uniquely on the chosen domain.
work through
Follow the decisions, not just the symbols
For f(t)=3t+2, the 3 is a rate and 2 is an initial value. At t=4, calculate 3×4+2=14. On a graph, the same decision appears as moving three vertical units for every one horizontal unit from a starting height of two.
experiment
Change one thing and watch the model answer
Change the rate and initial value in a linear rule. Predict which visual feature moves before checking: steepness belongs to the rate; vertical placement belongs to the initial value.
retrieve
Close the page and reconstruct it
Answer before opening the explanation. Retrieval is evidence only when the answer is produced without a hint.
Can two different inputs of a function produce the same output?
Yes. A function requires one output per input; it does not require different inputs to have different outputs.
transfer
Move the idea into a new setting
Describe the function connecting image width to height for a photograph whose aspect ratio must stay 3:2, using words, a table, and a formula.
reflect
Leave with a diagnostic habit
When translating representations, name the input, output, allowed domain, and invariant relationship. Symbols become easier once those choices are explicit.
Source record
Follow the idea back.
- Reference
- Algebra and Trigonometry 2e
- Publisher
- OpenStax
- License
- CC BY 4.0
- Accessed
- 2026-08-12
- URL
- https://openstax.org/details/books/algebra-and-trigonometry-2e