Linear/Quadratic/Cubic: domain = R (all real numbers)
Rational functions: domain = x where denominator ≠ 0
Square root functions: domain = x where radicand ≥ 0
Logarithmic functions: domain = x where argument > 0
Domain is the foundation of function analysis. Finding the domain requires considering all constraints: denominator ≠ 0, radicand ≥ 0 for even roots, argument > 0 for logs, and more.
⚠When multiple constraints exist simultaneously, the domain is the intersection of all constraint intervals. For example, √(x-1) + 1/(x-3) requires both x≥1 and x≠3.
What Is the Domain of a Function?
The domain of a function is the set of all possible input values (x) for which the function is defined. Finding the domain is the first step in studying function properties, with different function types having different constraints.
Set Concept
The domain is the set of valid x values for a function, one of the three essential elements (domain, range, mapping rule).
Constraints
Denominator ≠ 0 (rational), radicand ≥ 0 (even roots), argument > 0 (logs) are the three basic domain constraints.
Common Types
Linear and quadratic: R; rational: exclude denominator zeros; square root: radicand ≥ 0; log: argument > 0.
Intersection Method
Find the domain interval for each constraint separately, then take the intersection. When multiple constraints coexist, consider all conditions.
Teaching Example: Find domain of f(x) = √(x-1) / (x-3).
1. Square root requires: x - 1 ≥ 0 → x ≥ 1
2. Rational requires: x - 3 ≠ 0 → x ≠ 3
3. Intersection: x ≥ 1 and x ≠ 3 → domain = [1, 3) ∪ (3, +inf)
Applications
High School MathFunction PropertiesInequalitiesComposite FunctionsExam Prep
Frequently Asked Questions
What is the domain of a function?▼
The domain is the set of all possible input values (x) for which the function expression is defined. Different function types have different domain restrictions.
How do you find the domain of a rational function?▼
All x where denominator ≠ 0. Solve denominator≠0. For f(x)=1/(x-2): x-2≠0 → x≠2, domain = {x|x≠2}.
How do you find the domain of a square root function?▼
Radicand ≥ 0. For f(x)=√(x-3): x-3≥0 → x≥3, domain = [3,+inf). For quadratic radicands, solve a quadratic inequality.
What are the domains of linear and quadratic functions?▼
Both have domain = all real numbers R, because any real input yields a meaningful output.
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