Functions • Topic 3 of 4

Domain & Range

The domain of a function is the set of inputs that are allowed; the range is the set of outputs that actually come out. On the GRE two rules generate almost every domain restriction. First, a denominator can never be zero, so for f(x) = 1/(x − 4) the domain excludes x = 4. Second, the quantity under an even root (a square root) cannot be negative, so for f(x) = √(x − 2) you need x − 2 ≥ 0, i.e. x ≥ 2. Range questions lean on the shape of the rule: a squared term is always ≥ 0, so f(x) = x² + 5 has a minimum output of 5 and range "all values ≥ 5"; an absolute value behaves the same way. The exam rarely asks for domain or range in isolation — it hides the idea inside "for how many integer values of x is f(x) defined?" or a Quantitative Comparison about the smallest possible output. Check the denominator and the radicand first; that is where the restriction lives.

The square-root curve y = root(x minus 1) with its domain shaded on the x-axis and range on the y-axisDomain (inputs) & Range (outputs)xy-112345678910-11234Oy = √(x − 1)domain: x ≥ 1range: y ≥ 0

✅ Solved examples

1. Find the domain of f(x) = 1 / (x − 7).
The denominator cannot be zero: x − 7 ≠ 0, so x ≠ 7. Domain: all real numbers except 7.
2. Find the domain of f(x) = √(x − 5).
The radicand must be ≥ 0: x − 5 ≥ 0, so x ≥ 5. Domain: all x with x ≥ 5.
3. Find the range of f(x) = x² + 4.
x² ≥ 0 for every real x, so the smallest value of x² + 4 is 0 + 4 = 4. Range: all values ≥ 4.
4. Find the range of f(x) = |x − 3| − 2.
|x − 3| ≥ 0, so its smallest value is 0 (at x = 3), giving f = −2. Range: all values ≥ −2.

✏️ Practice — try these, take hints as needed

1. Find the domain of f(x) = 5 / (x + 2).
Denominator ≠ 0.
x + 2 ≠ 0.
Solve.
All real x except x = −2
2. Find the domain of f(x) = √(2x − 6).
Radicand ≥ 0.
2x − 6 ≥ 0.
2x ≥ 6.
x ≥ 3
3. Find the range of f(x) = x² − 9.
x² ≥ 0.
Smallest x² is 0.
0 − 9.
All values ≥ −9
4. Find the range of f(x) = 3 − |x|.
|x| ≥ 0 so −|x| ≤ 0.
Largest value when |x| = 0.
3 − 0 is the maximum.
All values ≤ 3
5. For how many integers x from 1 to 10 is f(x) = 1/(x − 5) undefined?
Undefined where denominator is 0.
x − 5 = 0.
Count how many such x in the range.
Exactly one (x = 5)

📝 Topic test — 8 questions

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