Functions: Domain, Range, and Notation You Actually Need
Master the language of functions — what domain and range really mean, how to read function notation, and why piecewise functions trip students up.
The function mindset
A function is a rule that takes an input and produces exactly one output. The set of allowed inputs is the domain; the set of resulting outputs is the range. Every Pre-Calculus problem touching functions assumes you can identify both.
Reading function notation
If f(x) = 2x + 3, then f(5) means "evaluate f when x = 5" → 2(5) + 3 = 13. The notation f(x + h) is NOT f(x) + f(h) — it means substitute the whole expression (x + h) for x.
Students often write f(x + 3) = f(x) + 3. This is only true if f is linear. For f(x) = x², we have f(x+3) = (x+3)² ≠ x² + 3.
Finding the domain
Two restrictions you always check:
- No division by zero: denominator ≠ 0
- No negative under even roots: radicand ≥ 0 for √, ⁴√, etc.
- No logs of zero or negative: argument > 0 for log/ln
For f(x) = √(x − 4) / (x − 9), both apply: x − 4 ≥ 0 AND x − 9 ≠ 0 → domain is [4, 9) ∪ (9, ∞).
Finding the range (harder than domain)
Strategies depending on function type:
- Squares + constant: x² ≥ 0 always; so x² + 5 ≥ 5; range [5, ∞).
- From the graph: project the curve onto the y-axis.
- Rational with same-degree: range = all reals except the horizontal asymptote value.
Check yourself
f(x) = 1/(x² − 16)?Practice with real CBE questions
Try domain and function notation questions in our Pre-Calc Sem A practice — start Sem A practice.