Table of Domain and Range of Common Functions

A table of domain and range of common and useful functions is presented.
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Algebraic Functions

Function Domain Range
f(x) = x (-∞ , + ∞) (-∞ , + ∞)
f(x) = x 2 (-∞ , + ∞) [0 , + ∞)
f(x) = x3 (-∞ , + ∞) (-∞ , + ∞)
f(x) = xn , n even (-∞ , + ∞) [0 , + ∞)
f(x) = xn , n odd (-∞ , + ∞) (-∞ , + ∞)
f(x) = | x | (-∞ , + ∞) [0 , + ∞)
f(x) = Square root ( x ) [0 , + ∞) [0 , + ∞)
f(x) = Cube root ( x ) (-∞ , + ∞) (-∞ , + ∞)

Trigonometric Functions

Function Domain Range
f(x) = sin ( x ) (-∞ , + ∞) [-1 , 1]
f(x) = cos ( x ) (-∞ , + ∞) [-1 , 1]
f(x) = tan ( x ) All real numbers except π/2 + n*π (-in , + ∞)
f(x) = sec ( x ) All real numbers except π/2 + n*π (-∞ , -1] U [1 , + ∞)
f(x) = csc ( x ) All real numbers except n*π (-∞ , -1] U [1 , + ∞)
f(x) = cot ( x ) All real numbers except n*π (-∞ , + ∞)

Inverse Trigonometric Functions

Function Domain Range
f(x) = sin-1( x ) [-1 , 1] [-π/2 , π/2]
f(x) = cos-1( x ) [-1 , 1] [0 , π]
f(x) = tan-1( x ) (-∞ , + ∞) (-π/2 , + π/2)
f(x) = sec-1( x ) (-∞ , -1] U [1 , + ∞) [0 , π/2) U [π , 3 π/2)
f(x) = csc-1( x ) (-∞ , -1] U [1 , + ∞) (-π , -π/2] U (0 , π/2]
f(x) = cot-1( x ) (-∞ , + ∞) (0 , π)

Logarithmic and Exponential Functions

Function Domain Range
f(x) = a x (-∞ , + ∞) (0 , +∞)
f(x) = Log a ( x ) (0 , + ∞) (-∞ , + ∞)
f(x) = a x + k , k constant (-∞ , + ∞) (k , + ∞)
f(x) = Log a ( x - k) , k constant (k , + ∞) (-∞ , + ∞)

Hyperbolic Functions

Function Domain Range
sinh(x) = (e x - e -x) / 2 (-∞ , + ∞) (-∞ , + ∞)
cosh(x) = (e x + e -x) / 2 (-∞ , + ∞) [1 , + ∞)
tanh(x) = (e x - e -x) / (e x + e -x) (-∞ , + ∞) (-1 , + 1)
coth(x) = (e x + e -x) / (e x - e -x) (-∞ , 0) U (0 , + ∞) (-∞ , - 1) U (1 , +∞)
sech(x) = 2 / (e x + e -x) (-∞ , + ∞) (0 , 1)
sech(x) = 2 / (e x - e -x) (-∞ , 0) U (0 , +∞) (-∞ , 0) U (0 , +∞)

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