Understanding how to express the domain and range of a function using interval notation is a fundamental skill in algebra, precalculus, and calculus. In practice, this notation provides a concise, standardized way to describe sets of numbers—specifically, which input values (domain) a function can accept and which output values (range) it can produce. Mastering this language allows you to communicate mathematical ideas with precision, whether you are analyzing a simple quadratic equation or a complex piecewise function.
The Building Blocks: Symbols and Their Meanings
Before writing intervals, you must internalize the specific symbols used. Interval notation relies heavily on brackets and parentheses to indicate whether endpoints are included or excluded.
Parentheses: ( ) — Exclusive Endpoints Parentheses represent open intervals. They indicate that the endpoint value is not part of the set. Think of an open parenthesis as an open circle on a number line.
- Inequality equivalent: $x < a$ or $x > b$
- Example: $(2, 5)$ means all numbers strictly between 2 and 5. The values 2 and 5 themselves are excluded.
Brackets: [ ] — Inclusive Endpoints Brackets represent closed intervals. They indicate that the endpoint value is part of the set. Visualize a closed (filled-in) circle on a number line Less friction, more output..
- Inequality equivalent: $x \le a$ or $x \ge b$
- Example: $[2, 5]$ means all numbers from 2 to 5, including 2 and 5.
Infinity Symbols: $\infty$ and $-\infty$ — Unbounded Intervals Infinity is not a number; it is a concept representing "continues forever." Because you cannot reach or include infinity, you must always use a parenthesis next to $\infty$ or $-\infty$. A bracket next to infinity is mathematically incorrect.
- Example: $[3, \infty)$ means all numbers greater than or equal to 3.
- Example: $(-\infty, -1)$ means all numbers strictly less than -1.
The Union Symbol: $\cup$ — Combining Disjoint Sets When a domain or range consists of two or more separate intervals that do not overlap, you join them with the union symbol ($\cup$) No workaround needed..
- Example: $(-\infty, -2) \cup (2, \infty)$ describes all numbers less than -2 or greater than 2.
Step-by-Step Guide to Finding and Writing the Domain
The domain is the set of all permissible inputs ($x$-values). To write it in interval notation, follow this systematic process.
1. Identify Restrictions
Look at the function definition and hunt for "illegal" mathematical operations:
- Denominators cannot be zero. Set the denominator $\neq 0$ and solve for $x$.
- Radicands of even roots (square roots, fourth roots, etc.) must be non-negative. Set the expression inside the radical $\ge 0$ and solve.
- Arguments of logarithmic functions must be positive. Set the inside of the log ${content}gt; 0$ and solve.
- Real-world context constraints. If $x$ represents "number of items," the domain is restricted to non-negative integers (often written as $[0, \infty)$ in continuous models).
2. Solve the Inequalities
Convert the restrictions into simple inequality statements (e.g., $x \ge -3$, $x \neq 5$).
3. Plot on a Number Line (Mental or Physical)
Visualizing the allowed values prevents errors.
- Use closed circles $[\ ]$ for $\le$ or $\ge$.
- Use open circles $(\ )$ for ${content}lt;$ or ${content}gt;$ (and for $\neq$).
- Shade the allowed regions.
4. Translate to Interval Notation
Read the number line from left to right (negative infinity to positive infinity) Most people skip this — try not to..
- Write the leftmost bound first.
- Choose the correct bracket/parenthesis based on the circle type.
- Write the rightmost bound second.
- Use $\cup$ if there are gaps in the shading.
5. Domain Examples
Example A: Rational Function $f(x) = \frac{1}{x-4}$
- Restriction: Denominator $\neq 0 \rightarrow x - 4 \neq 0 \rightarrow x \neq 4$.
- Number line: All real numbers allowed except a hole at 4.
- Domain: $(-\infty, 4) \cup (4, \infty)$
Example B: Even Root Function $g(x) = \sqrt{x + 6}$
- Restriction: Radicand $\ge 0 \rightarrow x + 6 \ge 0 \rightarrow x \ge -6$.
- Number line: Closed circle at -6, shading to the right forever.
- Domain: $[-6, \infty)$
Example C: Rational with Even Root $h(x) = \frac{\sqrt{x-1}}{x-3}$
- Restriction 1 (Root): $x - 1 \ge 0 \rightarrow x \ge 1$.
- Restriction 2 (Denominator): $x - 3 \neq 0 \rightarrow x \neq 3$.
- Combined: $x \ge 1$ AND $x \neq 3$.
- Number line: Closed circle at 1, shading right, but an open hole at 3.
- Domain: $[1, 3) \cup (3, \infty)$
Step-by-Step Guide to Finding and Writing the Range
The range is the set of all possible outputs ($y$-values or $f(x)$ values). This is often trickier than the domain because it requires understanding the function's behavior—its peaks, valleys, asymptotes, and end behavior Worth knowing..
1. Analyze the Function Type
Different function families have predictable range patterns:
- Linear ($mx+b$): Range is almost always $(-\infty, \infty)$ (unless domain is restricted).
- Quadratic ($ax^2+bx+c$): Parabolas have a vertex $(h, k)$.
- If $a > 0$ (opens up): Range is $[k, \infty)$.
- If $a < 0$ (opens down): Range is $(-\infty, k]$.
- Even Root ($\sqrt{x}$): Output is always $\ge 0$. Range: $[0, \infty)$ (plus vertical shifts).
- Logarithmic ($\log x$): Range is always $(-\infty, \infty)$.
- Rational: Look for horizontal asymptotes and gaps in $y$-values.
2. Use Calculus or Algebra to Find Extrema
For polynomials or continuous functions on a closed interval, find critical points (derivative = 0 or undefined) to locate absolute minimums and maximums But it adds up..
3. Check Asymptotic Behavior
Does the function approach a value but never touch it? That value gets a parenthesis.
- $f(x) = \frac{1}{x}$ approaches 0 but never equals 0. Range: $(-\infty, 0) \cup (0, \infty)$.
4. Graph It (The Ultimate Check)
If you can sketch the graph accurately, the range is simply the projection of the graph onto the $y$-axis. Look at the lowest $y$-value reached (inclusive?) and the highest $y$-value reached (inclusive?).
5. Range Examples
Example A: Quadratic (Vertex Form) $f(x) = -2(x - 1)^2 +
