2.1 Functions
Evaluate a function at a given input.
Before you solve
Ideas in this lesson
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01Recognize functions and identify inputs and outputsWorked steps · five questions02Translate rules and machine diagramsWorked steps · five questions03Evaluate numerical inputs and tablesWorked steps · five questions04Evaluate symbolic inputs and distinguish operationsWorked steps · five questions05Evaluate piecewise rules at boundariesWorked steps · five questions06Calculate and interpret net changeWorked steps · five questions07Simplify difference quotients with restrictionsWorked steps · five questions08Find polynomial domains and restricted rangesWorked steps · five questions09Find rational-function domainsWorked steps · five questions10Find domains of even and odd rootsWorked steps · five questions11Combine domain restrictionsWorked steps · five questions12Connect verbal, formula, table, and graph representationsWorked steps · five questions13Evaluate and interpret physical modelsWorked steps · five questions14Build contextual piecewise cost rulesWorked steps · five questions15Create qualitative and data-based graphsWorked steps · five questions16Reason about unusual functions and model limitationsWorked steps · five questions
Lesson brief and rules
2.1 Functions
A function assigns exactly one output to each input in its domain. To evaluate , substitute a for every .
Remember
f(x) means the output of f at x; it does not mean f multiplied by x.
f(x) means the output of f at x; it does not mean f multiplied by x.
Worked example
01 / A relation consists of the pairs . Is it a function, and what is its range?
02 / Each input occurs once, so each has exactly one output.
03 / Repeated outputs are permitted.
04 / The range lists each attained output once: and .
Worked example 2 · Recognize functions and identify inputs and outputs
A machine sends input to both and . It sends each other input to one output. Is the complete relation a function?
Inspect the arrows leaving input .
There are two distinct assigned outputs.
A function requires exactly one output for every domain element.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.