What is the difference between domain and range?

Domain is the set of all possible input values (x-values) for which the function is defined. Range is the set of all possible output values (y-values) the function produces. Domain is about what you can plug in; range is about what you get out.

How do I write domain in interval notation?

Use parentheses ( ) for excluded endpoints and brackets [ ] for included endpoints. Use ∪ to join separate intervals. For example, the domain of f(x)=1/x is (-∞,0) ∪ (0,∞).

Does a rational function always exclude the horizontal asymptote from the range?

No. The range excludes the horizontal asymptote value only if the function never reaches that y-value. For f(x)=x/(x²+1), the function crosses y=0, so 0 is in the range. Always check by solving f(x)=asymptote value.

What is the domain of √(x²-4)?

The radicand must be ≥ 0, so x²-4 ≥ 0. This gives x ≤ -2 or x ≥ 2. In interval notation, the domain is (-∞,-2] ∪ [2,∞). Common mistake: forgetting the negative interval.

How do transformations affect domain and range?

Horizontal shifts (inside the function) shift the domain. Vertical shifts (+k outside) shift the range. Vertical stretches (multiply by a) stretch the range. For sine, f(x)=a sin(x)+d has range [-|a|+d, |a|+d].

What is the domain of tangent?

All real numbers except odd multiples of π/2, where the function has vertical asymptotes. In interval notation, it is ... ∪ (-3π/2, -π/2) ∪ (-π/2, π/2) ∪ (π/2, 3π/2) ∪ ...

What is the range of arcsin(x)?

The range of arcsin(x) is [-π/2, π/2]. Its domain is [-1, 1]. The domain of the inverse function equals the range of the original function, and vice versa.

Domain and Range of Parent Functions: Cheat Sheet

Domain and Range of Parent Functions: The 30-Second Cheat Sheet

You have a test in an hour. You need the domain and range of parent functions in a form you can scan, not a textbook chapter. This table is that form. It covers every parent function you will see in Algebra 1 through Precalculus, with the rule, a graph thumbnail, and the interval-notation answer. Bookmark or print the page. The domain and range of parent functions is the one reference you cannot afford to search for during a timed exam.

The Master Table: Function, Graph, Domain, Range

Each row gives the parent function, a mental picture of its graph, and the domain and range in interval notation. The notation uses brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints. A union symbol ∪ connects separate intervals.

Parent Functions: Domain and Range Chart

Function	Graph Shape	Domain	Range
Linear: f(x)=x	Straight line through origin	(-∞, ∞)	(-∞, ∞)
Quadratic: f(x)=x²	Parabola opening up, vertex at (0,0)	(-∞, ∞)	[0, ∞)
Cubic: f(x)=x³	S-shaped curve through origin	(-∞, ∞)	(-∞, ∞)
Square Root: f(x)=√x	Half-parabola starting at (0,0)	[0, ∞)	[0, ∞)
Cube Root: f(x)=∛x	S-shaped through origin	(-∞, ∞)	(-∞, ∞)
Absolute Value: f(x)=\|x\|	V-shaped, vertex at (0,0)	(-∞, ∞)	[0, ∞)
Rational: f(x)=1/x	Two branches, asymptotes at x=0 and y=0	(-∞, 0) ∪ (0, ∞)	(-∞, 0) ∪ (0, ∞)
Exponential: f(x)=bˣ (b>1)	Rises left to right, asymptote y=0	(-∞, ∞)	(0, ∞)
Logarithmic: f(x)=log_b(x) (b>1)	Rises right, asymptote x=0	(0, ∞)	(-∞, ∞)
Sine: f(x)=sin(x)	Wave oscillating between -1 and 1	(-∞, ∞)	[-1, 1]
Cosine: f(x)=cos(x)	Wave starting at 1	(-∞, ∞)	[-1, 1]
Tangent: f(x)=tan(x)	Repeating curves, asymptotes at π/2 + nπ	(-∞, ∞) except π/2 + nπ	(-∞, ∞)
Arcsin: f(x)=arcsin(x)	S-shaped, restricted	[-1, 1]	[-π/2, π/2]
Arccos: f(x)=arccos(x)	Decreasing, restricted	[-1, 1]	[0, π]
Arctan: f(x)=arctan(x)	S-shaped, asymptotes at y=±π/2	(-∞, ∞)	(-π/2, π/2)

Domain Restrictions Rules: The Three Exceptions

Three rules cover nearly every restriction you will see. Everything else has domain all real numbers.

Denominator Cannot Be Zero

For rational functions, set the denominator equal to zero and exclude those x-values. For f(x)=1/(x-2), exclude x=2. Each excluded value creates a vertical asymptote or a hole. A vertical asymptote occurs at a zero of the denominator that is not also a zero of the numerator. A hole occurs when a factor cancels; the x-value is still excluded.

Even-Index Radicand Must Be ≥ 0

For square roots, fourth roots, and any even root, the radicand must be non-negative. Solve the inequality. For f(x)=√(x-3), the domain is x≥3. For f(x)=√(x²-4), solve x²-4≥0 to get x≤-2 or x≥2, written as (-∞,-2] ∪ [2,∞). This is a common failure: students forget the negative interval. For cube roots and other odd roots, the radicand can be any real number.

Logarithm Argument Must Be Positive

For logarithmic functions, the argument of the log must be greater than zero. For f(x)=log₂(x+1), the domain is x>-1. The argument cannot be zero or negative.

Parent Functions Domain and Range Rules: How Transformations Shift the Answer

A parent function can be shifted, stretched, or reflected. Each transformation changes the domain and range in a predictable way.

Horizontal and Vertical Shifts

A horizontal shift (inside the function, like f(x-h)) moves the domain. For f(x)=√(x-2)+3, the domain shifts from [0,∞) to [2,∞). A vertical shift (+k outside) moves the range. For the same function, the range shifts from [0,∞) to [3,∞).

Stretches and Reflections

A vertical stretch (multiply by a) changes the range. For sine, f(x)=a sin(x), the range becomes [-|a|, |a|]. For exponential, f(x)=a·bˣ+k, the range is (k,∞) if a>0, (-∞,k) if a<0. The base bˣ is always positive, so the output is just that positive value shifted by k.

A reflection across the x-axis (a negative a) flips the range direction. For absolute value, f(x)=-|x|+5 has range (-∞,5]. For quadratic, f(x)=-(x-1)²+4 has range (-∞,4]. The vertex y-coordinate becomes the maximum instead of the minimum.

The most common mistake is ignoring the vertical shift. A student writes the range of f(x)=2ˣ+3 as (0,∞) instead of (3,∞). The shift is the asymptote.

Domain and Range Rules: Special Cases and Crossings

A rational function does not always exclude its horizontal asymptote from the range. For f(x)=x/(x²+1), the horizontal asymptote is y=0, but the function crosses it at x=0, so 0 is in the range. The range is all real numbers except the asymptote value only if the function never reaches it. To check, set the function equal to the asymptote value and solve. If there is a real solution, the value is in the range.

For piecewise functions, the domain is the union of the intervals where each piece is defined. The range is found by evaluating each piece over its interval and taking the union of all resulting y-values. Each piece boundary must be checked for open vs. closed endpoints.

For composite functions, the domain of the outer function is restricted by the range of the inner function. The domain of f(g(x)) is all x such that g(x) is in the domain of f.

Who Needs a Parent-Function Reference

This cheat sheet is for Algebra 1 students learning the definition, Algebra 2 students tackling rational and radical functions, and Precalculus students handling piecewise and inverse trigonometric functions. It is also for teachers and tutors who need a reliable reference for lesson planning.

Anyone looking for calculus-level continuity analysis or epsilon-delta proofs should use a calculus textbook. Students who cannot evaluate a function at a given x-value should first master function notation from OpenStax College Algebra 2e, Section 3.1.

Printable Version: Print Stylesheet Instructions

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