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.
Definition of the Range of a Function
The range of f is the set of all values that the function
takes when x takes values in the domain.
Example 1: Find the domain of function f
defined by
f (x) = 1 / ( x  1)
Solution to Example 1
 x can take any real number except 1 since x = 1 would make the denominator equal to zero and the division by zero is not allowed in mathematics. Hence the domain in interval notation is given by
(infinity , 1) U (1 , +infinity)
Matched Problem 1: Find the domain of
function f defined by
f (x) = 1 / ( x + 3)
Answers to matched problems 1,2,3 and 4
Example 2: Find the domain of function f
defined by
f (x) = sqrt (2x  8)
Solution to Example 2
 The expression defining function f
contains a square root. The expression under the radical has to satisfy the
condition
2x  8 >= 0 for the function to take real
values.
 Solve the above linear inequality
x >= 4
 The domain, in interval notation, is given
by
[4 , +infinity)
Matched Problem 2: Find the domain of function f defined by:
f (x) = sqrt (x + 9)
Example 3: Find the domain of function f
defined by:
f (x) = sqrt( x) / [(x  3) (x + 5)]
Solution to Example 3
 The expression defining function f
contains a square root. The expression under the radical has to satisfy the
condition
x >= 0
 Which is equivalent to
x <= 0
 The denominator must not be zero, hence x
not equal to 3 and x not equal to 5.
 The domain of f is given by
(infinity , 5) U ( 5 , 0]
Matched Problem 3: Find the domain of function f defined by:
f (x) = sqrt( x + 2) / [(x + 1)
(x + 9)]
Example 4: Find the range of function f
defined by:
f (x) = x^{ 2}  2
Solution to Example 4
 The domain of this function is the set of
all real numbers. The range is the set of values that f(x) takes as x varies. If x is a real number, x^{2} is either positive or zero. Hence we can write the following:
x^{ 2} >= 0
 Subtract 2 to both sides to obtain
x^{ 2}  2>= 2
 The last inequality indicates that x^{2} 2 takes all values
greater that or equal to 2. The range of f is given by
[ 2 , +infinity)
 A graph of f also helps in interpreting the range of a function.
Below is shown the graph of function f given above. Note the lowest point in the graph
has a y (= f (x) ) value of 2.
Matched Problem 4: Find the range of
function f defined by:
f (x) = x^{ 2} + 3
More Find the range of functions,
find the domain of a function and mathematics tutorials and problems.
