Domain and Range — SAT Math Explained
The domain is the set of all valid input values (x-values) for a function. The range is the set of all possible output values (y-values) that the function can produce.
The Core Idea
Not every function can accept every input. Division by zero, square roots of negatives, and logarithms of non-positives create restrictions. Domain tells you what's allowed in; range tells you what can come out.
Key Vocabulary
All valid x-values (inputs)
All resulting y-values (outputs)
A value excluded from the domain due to mathematical impossibility
[a, b] means a ≤ x ≤ b; (a, b) means a < x < b; use ∪ for union of intervals
Common Domain Restrictions
Radicand must be ≥ 0 (can't take the square root of a negative)
Denominator ≠ 0 (can't divide by zero)
Argument must be > 0
Linear and polynomial functions have domain = all real numbers
Finding Domain
1. Look for square roots: set radicand ≥ 0 and solve
2. Look for denominators: set denominator ≠ 0 and solve
3. Look for logarithms: set argument > 0
4. Combine restrictions: use 'and' (intersection) if both apply
Finding Range
From a graph: identify the minimum and maximum y-values the graph reaches
From the equation: analyze the behavior — for y = x², the minimum output is 0 so range is [0, ∞)
For linear functions: range = all real numbers (unless domain is restricted)
Notation
Written in interval notation [min, max] with square brackets for included endpoints and parentheses for excluded ones; ∞ always gets a parenthesis
Common Errors to Avoid
Confusing domain (x-values) with range (y-values)
Forgetting to solve the square root restriction as ≥ 0, not just > 0
Not writing the domain/range in proper interval notation
Practice: Domain and Range
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What is the domain of f(x) = 2x + 5?
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