Definition of the Domain of a Function
For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given
explicitly.
also
Step by Step Calculator to Find Domain of a Function
Example 1: Find the domain of function f
defined by
f (x) = log_{3}(x  1)
Solution to Example 1

f(x) can take real values if the argument of log_{3}(x  1) which is x  1 is positive. Hence the condition on the argument
x  1 > 0

Solve the above inequality for x to obtain the domain: x > 1 or in interval for (∞ , 1)
Matched Problem 1: Find the domain of
function f defined by
f (x) = log_{5}(3  x)
Example 2: Find the domain of function f
defined by
f (x) = log_{2}(x^{2} + 5)
Solution to Example 2

The argument of log_{2}(x^{2} + 5) which is x^{2} + 5 is always greater than zero and therefore positive. Hence the domain of the given function is given by the interval: (∞ , +∞)
Matched Problem 2: Find the domain of function f defined by:
f (x) = ln (3 + x^{4})
Example 3: Find the domain of function f
defined by:
f (x) = ln (9  x^{2})
Solution to Example 3

For ln (9  x^{2}) to be real, the argument of ln (9  x^{2}) which is 9  x^{2} must be positive. Hence the inequality
9  x^{2} > 0

The solution of the inequality 9  x^{2} > 0 is given by interval
(3 , 3)

The domain of the given function is given by the interval (3 , 3).
Matched Problem 3: Find the domain of function f defined by:
f(x) = log_{4} (16  x^{2})
Example 4: Find the domain of function f
defined by:
f (x) = log_{4} x  3
Solution to Example 4

The domain of this function is the set of
all values of x such that x  3 > 0. The expression x  3 is positive for all real values except for x = 3 which makes it zero. Hence the domain of the given function is the set of all real values except 3, which can be written in interval form as follows
(∞ , 3) U (3 , +∞)

or in inequality form as follows
x < 3 or x > 3
Matched Problem 4: Find the domain of
function f defined by:
f (x) = ln  x  6
Example 5: Find the domain of function f
defined by:
f (x) = ln (2 x^{2}  3x  5)
Solution to Example 5

The domain of this function is the set of
all values of x such that 2 x^{2}  3x  5 > 0. We need to solve the inequality
2 x^{2}  3x  5 > 0

Factor the expression on the left hand side of the inequality
(2x  5)(x + 1) > 0

Solve the above inequality to obtain the solution set as follows:
x < 1 or x > 5 / 2

The domain is given in inequality form as x < 1 or x > 5 / 2 and in interval form as follows:
(∞ , 1) U (5 / 2 , +∞)
Matched Problem 5: Find the domain of
function f defined by:
f (x) = log ( 3x^{2} + 4x  7 )
Answers to matched problems
 (∞ , 3)
 (∞ , ∞)
 (4 , 4)
 (∞ , 6) ∪ (6 , ∞)
 (∞ , 7/3) ∪ (1 , ∞)