How to Calculate the Domain of a Function
- 1). Check the function for when denominators are zero. Since you can't divide by the number zero, the function can't take such "input" values and return a single, meaningful "output" value.
For example, f(x)=5x/(1-x) has zero in the denominator when 1-x=0, i.e. when x=1. In this case, the domain of f(x) is all real numbers except zero. - 2). Check the function for when the square roots are negative.
For example, f(x)=√(1-x) has negatives under the square root sign when 1-x<0, i.e. when 1<x. In this case, the domain of f(x) is "all real numbers greater than or equal to 1." - 3). Check the function for when the arguments of logarithms are equal to or less than 0.
For example, f(x)=log(1-x) has that problem with 1-x≤0, i.e. when 1≤x. In this case, the domain of f(x) is "all real numbers less than 1." - 4). Check the function for trigonometric arguments that must be within a certain range.
For example, f(x)=arcsin x is the same as saying sin (f(x)) = x. Since sine returns only values from -1 to 1, then x can have values only from -1 to 1. The domain of f(x) is therefore from -1 to 1 only.
Source...