Table of Domain and Range of Common Functions

A table of domain and range of common and useful functions is presented.
Also a Step by Step Calculator to Find Domain of a Function and a Step by Step Calculator to Find Range of a Function are included in this website.

Algebraic Functions

Function Domain Range
\( f(x) = x \) \( (-\infty, +\infty) \) \( (-\infty, +\infty) \)
\( f(x) = x^2 \) \( (-\infty, +\infty) \) \( [0, +\infty) \)
\( f(x) = x^3 \) \( (-\infty, +\infty) \) \( (-\infty, +\infty) \)
\( f(x) = x^n \), \( n \) even \( (-\infty, +\infty) \) \( [0, +\infty) \)
\( f(x) = x^n \), \( n \) odd \( (-\infty, +\infty) \) \( (-\infty, +\infty) \)
\( f(x) = |x| \) \( (-\infty, +\infty) \) \( [0, +\infty) \)
\( f(x) = \sqrt{x} \) \( [0, +\infty) \) \( [0, +\infty) \)
\( f(x) = \sqrt[3]{x} \) \( (-\infty, +\infty) \) \( (-\infty, +\infty) \)

Trigonometric Functions

Function Domain Range
\( f(x) = \sin(x) \) \( (-\infty, +\infty) \) \( [-1, 1] \)
\( f(x) = \cos(x) \) \( (-\infty, +\infty) \) \( [-1, 1] \)
\( f(x) = \tan(x) \) All real numbers except \( \frac{\pi}{2} + n\pi \) \( (-\infty, +\infty) \)
\( f(x) = \sec(x) \) All real numbers except \( \frac{\pi}{2} + n\pi \) \( (-\infty, -1] \cup [1, +\infty) \)
\( f(x) = \csc(x) \) All real numbers except \( n\pi \) \( (-\infty, -1] \cup [1, +\infty) \)
\( f(x) = \cot(x) \) All real numbers except \( n\pi \) \( (-\infty, +\infty) \)

Inverse Trigonometric Functions

Function Domain Range
\( f(x) = \sin^{-1}(x) \) \( [-1, 1] \) \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
\( f(x) = \cos^{-1}(x) \) \( [-1, 1] \) \( [0, \pi] \)
\( f(x) = \tan^{-1}(x) \) \( (-\infty, +\infty) \) \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)
\( f(x) = \sec^{-1}(x) \) \( (-\infty, -1] \cup [1, +\infty) \) \( [0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2}) \)
\( f(x) = \csc^{-1}(x) \) \( (-\infty, -1] \cup [1, +\infty) \) \( (-\pi, -\frac{\pi}{2}] \cup (0, \frac{\pi}{2}] \)
\( f(x) = \cot^{-1}(x) \) \( (-\infty, +\infty) \) \( (0, \pi) \)

Logarithmic and Exponential Functions

Function Domain Range
\( f(x) = a^x \) \( (-\infty, +\infty) \) \( (0, +\infty) \)
\( f(x) = \log_a(x) \) \( (0, +\infty) \) \( (-\infty, +\infty) \)
\( f(x) = a^x + k \), \( k \) constant \( (-\infty, +\infty) \) \( (k, +\infty) \)
\( f(x) = \log_a(x - k) \), \( k \) constant \( (k, +\infty) \) \( (-\infty, +\infty) \)

Hyperbolic Functions

Function Domain Range
\( \sinh(x) = \frac{e^x - e^{-x}}{2} \) \( (-\infty, +\infty) \) \( (-\infty, +\infty) \)
\( \cosh(x) = \frac{e^x + e^{-x}}{2} \) \( (-\infty, +\infty) \) \( [1, +\infty) \)
\( \tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) \( (-\infty, +\infty) \) \( (-1, 1) \)
\( \coth(x) = \frac{e^x + e^{-x}}{e^x - e^{-x}} \) \( (-\infty, 0) \cup (0, +\infty) \) \( (-\infty, -1) \cup (1, +\infty) \)
\( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \) \( (-\infty, +\infty) \) \( (0, 1) \)
\( \operatorname{csch}(x) = \frac{2}{e^x - e^{-x}} \) \( (-\infty, 0) \cup (0, +\infty) \) \( (-\infty, 0) \cup (0, +\infty) \)


References and Links

Find domain and range of functions
Find the range of functions
find the domain of a function and mathematics tutorials and problems
Step by Step Calculator to Find Domain of a Function

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