Question 1

What is the domain of a function?

Accepted Answer

The domain is the set of all x-values for which the function f(x) is defined and produces a real number output. For example, f(x) = 1/x is defined for all real numbers except x = 0, so its domain is (−∞, 0) ∪ (0, ∞). Polynomial functions like f(x) = x² + 3x have domain (−∞, ∞) — every real number is valid. Identifying the domain is the first step in analyzing any function.

Question 2

What is the range of a function?

Accepted Answer

The range is the set of all output values f(x) as x runs over the domain. For f(x) = x², every output is non-negative, so the range is [0, ∞). For f(x) = sin(x), the output oscillates between −1 and 1, giving range [−1, 1]. Finding the range often requires solving f(x) = y for x and determining which y-values yield real solutions.

Question 3

How do you find the domain of a rational function?

Accepted Answer

Set the denominator equal to zero and solve for x. Exclude those x-values from the domain. For f(x) = (x+2)/(x²−4), set x²−4 = 0, giving x = ±2. The domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). If the numerator also equals zero at the same point, the function may have a removable discontinuity (hole) rather than a vertical asymptote, but that x-value is still excluded from the domain.

Question 4

How do you find the domain of a square root function?

Accepted Answer

The expression inside the square root (the radicand) must be greater than or equal to zero, because the square root of a negative number is not a real number. For f(x) = √(2x − 6), set 2x − 6 ≥ 0, giving x ≥ 3. The domain is [3, ∞). For nested radicals or radical fractions, apply both the non-negative radicand rule and the non-zero denominator rule simultaneously.

Question 5

What is interval notation and how do I use it?

Accepted Answer

Interval notation represents sets of numbers using parentheses and brackets. A parenthesis ( or ) means the endpoint is excluded (open interval); a bracket [ or ] means the endpoint is included (closed interval). Use ∞ or −∞ (always with parentheses, since infinity is not a number) to indicate the interval extends without bound. For example, the domain x > 3 is (3, ∞), while x ≥ 3 is [3, ∞). To express a domain with a gap, use the union symbol ∪, e.g., (−∞, −2) ∪ (−2, ∞).

Function Type	Example	Domain Restriction	Domain	Range
Polynomial	f(x) = x² + 3	None	(−∞, ∞)	Depends on degree
Linear	f(x) = 2x + 1	None	(−∞, ∞)	(−∞, ∞)
Rational 1/(x−a)	f(x) = 1/(x−3)	x ≠ 3	(−∞,3)∪(3,∞)	(−∞,0)∪(0,∞)
Square root √(x−a)	f(x) = √(x−2)	x ≥ 2	[2, ∞)	[0, ∞)
Logarithm log(x−a)	f(x) = log(x−1)	x > 1	(1, ∞)	(−∞, ∞)
Quadratic ax²+bx+c	f(x) = x²−4x+3	None	(−∞, ∞)	[−1, ∞)
Absolute value \|ax+b\|	f(x) = \|x−2\|	None	(−∞, ∞)	[0, ∞)
Even root ⁴√(f(x))	f(x) = ⁴√(x)	x ≥ 0	[0, ∞)	[0, ∞)
Natural log ln(x)	f(x) = ln(x)	x > 0	(0, ∞)	(−∞, ∞)
Sine / Cosine	f(x) = sin(x)	None	(−∞, ∞)	[−1, 1]

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