Domain and Range of Basic Functions

A table of domain and range of basic and useful functions. Note that in what follows -inf means - infinity and +inf means + infinity.
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Algebraic Functions

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

Trigonometric Functions

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

Inverse Trigonometric Functions

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

Logarithmic and Exponential Functions

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

Hyperbolic Functions

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

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