Understanding a function: inputs, outputs, and representations

A function assigns exactly one output to each input in its domain. Connect f(x), formulas, tables, graphs, and context.

1. Objectives and prerequisites

Prerequisites: sets and real numbers. You will learn to define a function, evaluate f(x), read outputs and preimages, connect representations, and choose a context-appropriate domain.

2. Definition: one input, one output

A function f from a set D to a target set assigns exactly one output f(x) to each x in D. Several inputs may share one output; one input cannot have two different outputs.

3. Domain, value, and preimage

The domain D contains allowed inputs. The value at x is f(x). A preimage of y is an x such that f(x)=y. Context can restrict the domain: a trip distance is not negative even if an algebraic formula accepts negative numbers.

4. Four representations of one relationship

The same function can be described verbally, by formula, table, or graph. Changing representation does not change the rule. The graph is the set of points (x,f(x)).

5. Worked example

For f(x)=2x+3 over the real numbers, f(2)=7, so (2,7) is on the graph. Solving f(x)=7 gives 2x+3=7, hence x=2.

6. Practice

6.1. Evaluate and read a function

Let f(x)=2x+3. Compute f(-2), f(0), and f(4), then find the input whose output is 7. Present the results in an input-output table.

Guided correction. f(-2)=-1, f(0)=3, f(4)=11, and the input for output 7 is x=2. The table clearly separates inputs and outputs.

6.2. Model a trip fare

A trip costs a €4 fixed charge plus €1.80 per kilometer. Define C(d), state a realistic domain, compute C(0), C(2), and C(5), plot the points, and explain 1.80 and 4.

Success criteria. C(d)=4+1.8d with d≥0 in the chosen model; C(0)=4, C(2)=7.6, C(5)=13. 1.80 is cost per km and 4 is the fixed charge.

6.3. Recognize a function and interpret values

A table maps -1→3, 0→1, 1→1, and 2→3. Decide whether it defines a function, give the output for 2 and the inputs whose output is 1. Explain why a relation containing both 0→1 and 0→4 would not be a function.

Success criteria. The justification applies the “one output per input” rule, reads the table correctly, and distinguishes this from multiple inputs sharing one output.

7. Summary

A function assigns exactly one output to each input in its domain. Formula, table, graph, and words are representations of the same relationship.

8. Curriculum sources

French Ministry of Education — 2026 Seconde mathematics curriculum, Functions section: official bulletin.
California Department of Education — F-IF.1 and F-IF.2: California Content Standards.

9. Revision note

Lesson prepared on 20 September 2026 from two official frameworks from different countries. Text and translations were prepared with AI; no measured learning gain is claimed.