How to Find the Domain and Range of a Function

To learn how to find domain and range manually, follow four steps: identify the type, list restrictions, solve and write the domain, then find the range. The method works for polynomial, rational, radical, logarithmic, exponential, trigonometric, and piecewise functions. Use interval notation and watch for common mistakes that cost points.

Look at the function and decide which family it belongs to. The family determines domain and range rules. A polynomial like f(x)=x²-4x+3 has domain all real numbers. A rational function like f(x)=(x+1)/(x-2) has denominator restrictions. A square root function like f(x)=√(x-3) requires a radicand ≥ 0. Logarithmic functions need the argument > 0. Exponential functions have no domain restrictions but a shifted range. Trigonometric functions have specific domain or range limits, especially inverse functions.

If the function combines types, each restriction applies. OpenStax Precalculus 2e, Section 3.1 (pages 199-214) defines function notation, and Section 3.2 (pages 215-230) covers domain and range per family. Paul's Online Math Notes Algebra section "The Definition of a Function" provides the same breakdown.

Three restrictions cover most homework problems. First, denominator zero: for any rational function, exclude x-values that make the denominator equal zero. A vertical asymptote occurs at a denominator zero that does not cancel; a hole occurs if the factor cancels. Second, even-index radicand: for square roots or fourth roots, set the radicand ≥ 0. For cube roots, no restriction applies. Third, logarithmic argument: for log base anything, the argument must be > 0. Exponential functions have no restrictions on the domain.

A function can have two or three restrictions at once. For example, f(x)=√(x+1)/(x²-4) has both a radicand and a denominator restriction. Paul's Online Math Notes "Domain and Range examples" works through combined cases.

Turn each restriction into an inequality or equation. For denominator zero: solve denominator = 0 and exclude those values. For radicand: solve radicand ≥ 0. For log argument: solve argument > 0. Write the solution in interval notation. Use parentheses ( ) for excluded endpoints and brackets [ ] for included endpoints. If the domain has two separate intervals, join them with the union symbol ∪.

Example: f(x)=1/(x-5) has denominator zero at x=5. Domain: (-∞, 5) ∪ (5, ∞). Example: g(x)=√(x+2) has radicand x+2 ≥ 0, so x ≥ -2. Domain: [-2, ∞). Example: h(x)=log(x-1) has argument x-1 > 0, so x > 1. Domain: (1, ∞).

Domain: (-∞, -2] ∪ [2, ∞). This is a common source of errors when students write "x≥2, x≤-2" without the union symbol.

The range is harder than the domain. Use these methods depending on function type:

Quadratic Functions

Find the vertex. For f(x)=ax²+bx+c, the x-coordinate is -b/(2a). Plug it in to get the y-coordinate. If a > 0, range is [vertex y, ∞). If a < 0, range is (-∞, vertex y]. OpenStax Precalculus 2e, Section 3.2 provides this method.

Rational Functions

Find the horizontal asymptote. For degrees equal, the asymptote is the ratio of leading coefficients. The range excludes that y-value only if the function never reaches it. Test by solving f(x)=asymptote value. If the equation has no solution, the asymptote value is excluded. If it has a solution, the value is included. This check is missing from most sources, including many textbooks. Paul's Online Math Notes "Domain and Range examples" shows the exclusion test.

Radical Functions

The range of √(stuff) starts at 0 (or the principal root value) and goes to ∞, unless the function has a vertical shift. For f(x)=√(x)+k, range is [k, ∞).

Logarithmic and Exponential Functions

Logarithmic functions have range all real numbers. Exponential functions have range (0, ∞) for the parent, or (k, ∞) with vertical shift. OpenStax Precalculus 2e, Chapter 6 covers both.

Trigonometric Functions

Sine and cosine have range [-1, 1] for the parent, modified by amplitude and vertical shift. Tangent has range all real numbers. Inverse sine has range [-π/2, π/2]; inverse cosine has range [0, π]; inverse tangent has range (-π/2, π/2). OpenStax Precalculus 2e, Chapter 7 and Chapter 8 provide these values.

Example 1: Polynomial

f(x)=x²-4x+3. Type: quadratic. Domain: all real numbers, (-∞, ∞). Vertex: x=-(-4)/(2*1)=2, f(2)=-1. Since a=1 > 0, range: [-1, ∞).

Example 2: Rational

f(x)=(x+1)/(x-2). Denominator zero at x=2, so domain: (-∞, 2) ∪ (2, ∞). Horizontal asymptote at y=1. Solve (x+1)/(x-2)=1, which gives 1=-2, no solution. Range excludes y=1. Range: (-∞, 1) ∪ (1, ∞).

Example 3: Square Root

f(x)=√(x-3).Range: [0, ∞).

Example 4: Square Root in Denominator

f(x)=1/√(x+2). Two restrictions: denominator cannot be zero, and radicand must be ≥ 0. The radicand gives x+2 ≥ 0, so x ≥ -2. The denominator zero occurs when √(x+2)=0, which happens at x=-2. Since x=-2 is excluded by the denominator, domain: (-2, ∞). Range: (0, ∞). The y-value goes to 0 as x → ∞ but never reaches 0.

Example 5: Log of a Quadratic

f(x)=log(x²-4). Argument x²-4 > 0.Domain: (-∞, -2) ∪ (2, ∞). Range: all real numbers, (-∞, ∞). The argument can approach 0 from above, making the log approach -∞, and can become arbitrarily large, making the log approach ∞.

Example 6: Combined Restrictions

f(x)=√(x-1)/(x²-9). Radicand: x-1 ≥ 0, so x ≥ 1. Denominator: x²-9 ≠ 0, so x ≠ 3 and x ≠ -3. Since -3 is already excluded by the radicand, combine: x ≥ 1 but x ≠ 3. Domain: [1, 3) ∪ (3, ∞). Range: find horizontal asymptote. Degrees: numerator degree 1/2, denominator degree 2, so HA at y=0.Range: (0, ∞).

Try these on your own, then check the answers.

1. f(x)=2x+3
2. f(x)=1/(x²-1)
3. f(x)=√(4-x)
4. f(x)=log(x+5)
5. f(x)=e^x + 2
6. f(x)=3/x

Answers:

1. Domain: (-∞, ∞); Range: (-∞, ∞)
2. Domain: (-∞, -1) ∪ (-1, 1) ∪ (1, ∞); Range: (-∞, -1] ∪ (0, ∞) (HA at y=0, but y=0 is never reached; check crossing: 1/(x²-1)=0 has no solution)
3. Domain: (-∞, 4]; Range: [0, ∞)
4. Domain: (-5, ∞); Range: (-∞, ∞)
5. Domain: (-∞, ∞); Range: (2, ∞)
6. Domain: (-∞, 0) ∪ (0, ∞); Range: (-∞, 0) ∪ (0, ∞)

• Forgetting denominator zero: Always check rational functions for zeros in the denominator. Even if the denominator looks like it cannot be zero, check.
• Assuming square root always gives positive only: The principal square root is non-negative, so range of √(stuff) is [0, ∞) unless shifted. Do not write (-∞, ∞).
• Mixing up domain and range: Domain is x-values, range is y-values. Write them in the correct order.
• Using wrong vertex coordinate: The range of a quadratic depends on the y-coordinate of the vertex, not the x-coordinate. Plug the x-coordinate into the function to get the y.
• Assuming range always excludes horizontal asymptote: Test whether the function ever reaches that y-value. Some rational functions cross the horizontal asymptote.
• Omitting the union symbol: Two separate intervals must be joined with ∪. Writing "x>2, x<5" is not correct interval notation.

If you cannot find the range using the methods above, sketch a graph. Plot key points, asymptotes, and vertex. A graphing tool can confirm your work, but never trust the calculator's window alone. The calculator shows only the portion on screen; excluded points may be invisible. Check the table of values for undefined entries. For piecewise functions, evaluate each piece at its boundary points. For rational functions with three or more exclusions, write each exclusion separately then take the union.

Function Type	Domain Rule	Range Rule	Example
Polynomial (any degree)	All real numbers	Odd degree: all reals; even degree: bounded one side	f(x)=x³-x: domain (-∞,∞), range (-∞,∞)
Rational (fraction of polynomials)	All reals except denominator zeros	All reals except horizontal asymptote value (if not crossed)	f(x)=1/x: domain (-∞,0)∪(0,∞), range (-∞,0)∪(0,∞)
Square root (even root)	Radicand ≥ 0	[0, ∞) or [k, ∞) with shift	f(x)=√x: domain [0,∞), range [0,∞)
Logarithmic	Argument > 0	All real numbers	f(x)=log(x): domain (0,∞), range (-∞,∞)
Exponential (base > 0, ≠1)	All real numbers	(0, ∞) or (k, ∞) with shift	f(x)=e^x: domain (-∞,∞), range (0,∞)
Sine and Cosine	All real numbers	[-1, 1] (modified by amplitude and shift)	f(x)=sin(x): domain (-∞,∞), range [-1,1]
Tangent	All reals except π/2 + kπ	All real numbers	f(x)=tan(x): domain excludes odd multiples of π/2, range (-∞,∞)
Inverse Sine	[-1, 1]	[-π/2, π/2]	f(x)=arcsin(x): domain [-1,1], range [-π/2,π/2]
Inverse Cosine	[-1, 1]	[0, π]	f(x)=arccos(x): domain [-1,1], range [0,π]
Inverse Tangent	All real numbers	(-π/2, π/2)	f(x)=arctan(x): domain (-∞,∞), range (-π/2,π/2)
Absolute Value	All real numbers	[k, ∞) or (-∞, k]	f(x)=|x|: domain (-∞,∞), range [0,∞)
Piecewise	Union of each piece's domain	Union of each piece's range	Check boundaries for open/closed endpoints