Domain & Range
The domain is the set of inputs the rule is allowed to accept; the range is the set of outputs it actually produces. CAT domain questions reduce to a short checklist: a square root forces the inside to be ≥ 0, a denominator forces it to be ≠ 0, and a logarithm forces its argument to be strictly > 0. Combine the conditions with AND — the domain is their intersection. Range is the harder half. The fast methods are: complete the square for a quadratic (a parabola has a clean minimum or maximum), bound expressions using the fact that a perfect square is ≥ 0, or solve y = f(x) for x and ask which y keep x real. For f(x) = 1/(x − a), every real except one value of y is reachable. Never plot points blindly — reason about what the rule can and cannot output.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Definitions and tests
| Function rule | each x in domain → exactly one f(x) |
|---|---|
| Even function | f(−x) = f(x) (graph symmetric about y-axis) |
| Odd function | f(−x) = −f(x) (graph symmetric about origin) |
| Periodic function | f(x + T) = f(x) for least T > 0 |
| Composite function | (f∘g)(x) = f(g(x)) |
| Inverse condition | f(f⁻¹(x)) = x and f⁻¹(f(x)) = x |
CAT power-tools
| Domain of √(g(x)) | need g(x) ≥ 0 |
|---|---|
| Domain of 1/g(x) | need g(x) ≠ 0 |
| One-one (injective) test | f(a) = f(b) ⇒ a = b |
| Onto (surjective) test | range = co-domain |
| Inverse of linear f(x)=ax+b | f⁻¹(x) = (x − b)/a |
| Self-inverse / involution | f(f(x)) = x (e.g. f(x)=1/x, a−x) |