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. Going deeper: what a function actually encodes

7.1. Relation or function?

A relation can connect objects without being a function. The decisive rule is about inputs: every input in the domain must have exactly one output. A table in which the same input is paired with two different outputs therefore fails immediately. On an ordinary graph y=f(x), the same idea becomes the vertical-line test: if one vertical line meets the graph more than once, a single x-value has several outputs.

The reverse is not required. Different inputs may share the same output. The square function is a simple example because (-2)² and 2² both equal 4.

7.2. Domain, codomain, and image

Three sets are often confused. The domain contains the allowed inputs. The codomain is the set in which outputs are declared to live. The image or range actually reached may be smaller. For f(x)=x² viewed from R to R, the image is [0,+∞). This distinction matters when finding preimages or asking whether a function can be inverted.

The domain is not always determined by algebra alone. For g(x)=1/(x-2), x=2 is excluded by the formula. In a model of time, mass, or distance, context can impose further restrictions even when the algebraic expression accepts more values.

7.3. Piecewise functions and changing representation

A rule may change on different parts of its domain. A tariff can be constant up to a threshold and affine afterwards; it is still one function provided each input gets exactly one output. A change in a piecewise formula must be reflected consistently in the table and graph.

Moving from words to a formula, table, and graph is also a model check. A formula is efficient for calculation, a table exposes selected values, a graph reveals global shape, and prose explains what the quantities and units mean. No representation is universally superior.

7.4. Reading graphs and common mistakes

Reading f(3)=5 means starting at x=3, reaching the graph, then reading y=5. Finding the preimages of 5 reverses that question: start from y=5 and identify every matching x-value. There may be none, one, or several.

Common mistakes include reading f(x) as f times x, treating codomain and image as identical, forgetting domain restrictions, believing several preimages violate the definition of a function, or inferring one unique formula from only a few table values. A function is first a rule on a specified domain, not a particular style of formula.

8. Summary

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

9. 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.

10. 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.