How to find the domain of any function algebraically — rational, radical, logarithmic, trig, and piecewise — with interval notation.
| Symbol | Meaning | Inequality | Example |
|---|---|---|---|
| [a, b] | Closed interval — both endpoints included | a ≤ x ≤ b | [2, 7]: includes 2 and 7 |
| (a, b) | Open interval — neither endpoint included | a < x < b | (2, 7): excludes 2 and 7 |
| [a, b) | Half-open — left included, right excluded | a ≤ x < b | [2, 7): includes 2, not 7 |
| (−∞, b] | All values up to and including b | x ≤ b | (−∞, 5]: all x ≤ 5 |
| (a, ∞) | All values greater than a | x > a | (3, ∞): all x > 3 |
| A ∪ B | Union — values in A OR in B | — | (−∞,0) ∪ (0,∞): all x except 0 |
| A ∩ B | Intersection — values in BOTH A and B | — | [1,5] ∩ [3,8] = [3,5] |
∞ and −∞ always use parentheses — infinity is a concept, not a number, so it is never included.
Domain is all real numbers — no restrictions
Example: f(x) = x³ − 5x + 2
Domain: (−∞, ∞)
Exclude values where the denominator = 0
Example: f(x) = (x+1)/(x²−4)
Domain: x²−4 = 0 → x = ±2 → (−∞,−2) ∪ (−2,2) ∪ (2,∞)
Radicand must be ≥ 0
Example: f(x) = √(3x − 6)
Domain: 3x−6 ≥ 0 → x ≥ 2 → [2, ∞)
Domain is all real numbers — no restrictions
Example: f(x) = ∛(x−5)
Domain: (−∞, ∞)
Argument must be strictly > 0
Example: f(x) = log(x² − 9)
Domain: x²−9 > 0 → x < −3 or x > 3 → (−∞,−3) ∪ (3,∞)
sin/cos: all reals. tan/sec: exclude x = π/2 + nπ. cot/csc: exclude x = nπ
Example: f(x) = tan(x)
Domain: x ≠ π/2 + nπ for any integer n
Example: f(x) = √(x+3) / (x² − x − 6)
Restriction 1 — Radicand ≥ 0:
x + 3 ≥ 0 → x ≥ −3 → [−3, ∞)
Restriction 2 — Denominator ≠ 0:
x² − x − 6 = 0 → (x−3)(x+2) = 0 → x = 3 or x = −2
Exclude x = 3 and x = −2
Combine restrictions:
Start with [−3, ∞), remove x = −2 and x = 3
Domain: [−3, −2) ∪ (−2, 3) ∪ (3, ∞)
Step 1: Find the domain of g(x) — call this D_g
Step 2: Find the domain of f(x) — call this D_f
Step 3: Find all x in D_g where g(x) is in D_f
Step 4: The domain of f(g(x)) is the intersection of the above
Example: f(x) = √x, g(x) = x − 4. Find domain of f(g(x)).
f(g(x)) = √(x − 4)
Domain of g: all reals. Domain of f: x ≥ 0.
Need g(x) ≥ 0: x − 4 ≥ 0 → x ≥ 4
Domain of f(g(x)): [4, ∞)
Use the vertex. If a > 0 (opens up), range is [k, ∞). If a < 0 (opens down), range is (−∞, k]. k = vertex y-value.
f(x) = x² − 4x + 7 = (x−2)² + 3 → range [3, ∞)
√ always outputs ≥ 0. Shift vertically by any constant outside the radical.
f(x) = √(x+3) − 1 → range [−1, ∞)
Check the horizontal asymptote — the range excludes that y-value (unless there's a hole at the asymptote). Solve y = f(x) for x to find excluded values.
f(x) = (x+1)/(x−2), HA y=1 → range: (−∞,1) ∪ (1,∞)
b^x > 0 always. Range is (0, ∞) for standard exponential. Shift vertically: y = b^x + k has range (k, ∞).
f(x) = 3^x − 2 → range (−2, ∞)
Start with all real numbers and remove values that cause problems: (1) Division by zero — set denominator ≠ 0; (2) Even roots of negatives — set radicand ≥ 0; (3) Logarithms — set argument > 0. For composition f(g(x)), first find domain of g, then restrict to values where g(x) is in the domain of f. Express the answer in interval notation.
Interval notation uses brackets and parentheses: [ ] means the endpoint IS included (closed), ( ) means the endpoint is NOT included (open). Use ∞ and −∞ with parentheses (never brackets, since infinity is not a number). Examples: [2, 5) means 2 ≤ x < 5; (−∞, 3] means x ≤ 3; (1, 3) ∪ (5, ∞) means x is between 1 and 3 OR greater than 5.
The range is all possible output values (y-values) of a function. Finding range is harder than finding domain — common strategies: (1) Graph the function and read off the y-values; (2) Solve y = f(x) for x and find what y values allow a real x solution; (3) For quadratics, use the vertex; (4) For rational functions, check for horizontal asymptotes and holes.
Interactive problems with step-by-step solutions and private tutoring — free to try.
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