Section 1.fn

Functions, Domain & Range

Domain and range questions appear on nearly every precalc final — expect at least 2–3 on yours. Every topic from here (polynomials, trig, logs) builds on knowing what a function is and what inputs are allowed.

⚡ Quick Summary

Think of a function like a vending machine: you press one button (the input) and get exactly one snack (the output). If pressing the same button sometimes gave you chips and sometimes gave you candy, that would be chaos — and it wouldn't be a function. The rule is simple: one input, one output, every time.

Domain of a Rational Function

f(x) = \frac{p(x)}{q(x)} \implies \text{exclude all } x \text{ where } q(x) = 0

Set the denominator equal to zero and solve — those x-values are NOT in the domain

Domain of a Square Root Function

f(x) = \sqrt{g(x)} \implies g(x) \geq 0

The expression under the radical must be zero or positive

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The domain is just the list of buttons that actually work on the machine — some inputs might cause errors. In math, two things break functions: dividing by zero (your calculator would crash) and taking the square root of a negative number (which doesn't give a real answer). The range is the collection of snacks that can actually come out — all the possible outputs.

We write domains and ranges using interval notation, which is just a shorthand for describing chunks of the number line. Parentheses $(\,)$ mean "not included" and brackets $[\,]$ mean "included." Infinity always gets a parenthesis because you can never actually reach it — it's a direction, not a destination.

When you see a new function, run a quick safety check: if there's a fraction, make sure the bottom never equals zero; if there's a square root, make sure the inside stays zero or positive. Handle both at the same time if the function has both. That's really all there is to finding the domain.

Key Formulas

Domain of a Rational Function

f(x) = \frac{p(x)}{q(x)} \implies \text{exclude all } x \text{ where } q(x) = 0

Set the denominator equal to zero and solve — those x-values are NOT in the domain

Domain of a Square Root Function

f(x) = \sqrt{g(x)} \implies g(x) \geq 0

The expression under the radical must be zero or positive

Interval Notation Quick Reference

(a, b) \;\text{open} \quad [a, b] \;\text{closed} \quad [a, b) \;\text{half-open} \quad (-\infty, \infty) \;\text{all reals}

Infinity always gets a parenthesis, never a bracket

Key Takeaways

✓A function assigns exactly one output to each input — the vertical line test checks this visually
✓Domain = all valid inputs; Range = all outputs that actually occur
✓Two domain killers: division by zero (denominator $= 0$ ) and square root of a negative (radicand $< 0$ )
✓Interval notation: parentheses $(\,)$ for excluded endpoints, brackets $[\,]$ for included — infinity always gets a parenthesis

⚠️ Common Mistakes

✗Writing the domain of $\sqrt{x - 3}$ as $x > 3$ instead of $x \geq 3$ — the square root of zero is perfectly fine, it's only negatives that break things
✗Forgetting to exclude BOTH values when the denominator factors into two pieces — e.g., $x^2 - 9 = 0$ gives $x = 3$ AND $x = -3$ , not just one of them
✗Confusing domain (valid inputs / $x$ -values) with range (outputs / $y$ -values) — domain is horizontal, range is vertical
✗Using a bracket instead of a parenthesis at infinity — $\infty$ always gets a parenthesis because you can never actually reach it

Next: Section 1.comp

1.comp Composition & Transformations