Processing math: 100%

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