Function Concepts • Topic 3 of 5

Domain

The domain of a function is the set of all input (x) values for which the function is defined. For most polynomials the domain is all real numbers, but two situations restrict it. A fraction is undefined where its denominator is zero, so those x-values are excluded — for f(x) = 1/(x − 4), x cannot be 4. A square root needs a non-negative inside, so for f(x) = √(x − 3) you need x − 3 ≥ 0, giving x ≥ 3. To find the domain, look for denominators that could be zero and radicals that could go negative, and exclude or restrict accordingly. The SAT focuses on these two cases.

A number line with a filled dot at 3 and the ray shaded to the right, showing x is greater than or equal to 3.Domain0123456Domain of √(x − 3): x ≥ 3

✅ Solved examples

1. What x is excluded from f(x) = 1/(x − 5)?
Denominator zero at x = 5, so x = 5 is excluded.
2. Find the domain of f(x) = √(x − 2).
Need x − 2 ≥ 0, so x ≥ 2.
3. What is the domain of f(x) = 3x + 1?
All real numbers (no restriction).
4. What x is excluded from f(x) = 1/x?
x = 0.

✏️ Practice — try these, take hints as needed

1. What x is excluded from f(x) = 1/(x − 7)?
Denominator ≠ 0.
x − 7 = 0.
—
x = 7.
2. Find the domain of f(x) = √(x − 5).
Inside ≥ 0.
x − 5 ≥ 0.
—
x ≥ 5.
3. What x is excluded from f(x) = 1/(x + 2)?
x + 2 = 0.
—
—
x = −2.
4. Find the domain of f(x) = √(x + 4).
x + 4 ≥ 0.
—
—
x ≥ −4.
5. What is the domain of f(x) = 2x − 9?
Any restrictions?
No denominator or root.
—
All real numbers.

📝 Topic test — 8 questions

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