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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, x2 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 x2- 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 on finding the domain of a function and mathematics tutorials and problems.
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