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♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥
DOMAIN AND RANGE OF A FUNCTION
Jump right to the exercises!
The domain of a function is the set of all its allowable inputs.
The domain convention states that if the domain of a function is not
specified,
then it is assumed to be the set of all real numbers for which the function is defined.
The range of a function is the set of all its outputs, as the inputs vary through
the entire domain.
The domain of a function f is denoted by
dom(f) .
The range of a function f is denoted by
ran(f) .
Since the domain and range are sets, correct set notation must be used when reporting them.
It may be helpful to review interval and list notation.
Remember that the symbol ℝ denotes the set of real numbers.
The domain of a function is usually quite easily determined from the formula for the function.
Numbers that cause division by zero must be excluded from the domain.
Anything inside an even root (square root, fourth root, etc.) must be greater than or equal to zero.
The range of a function is usually more difficult to determine from a formula.
Often, it is much easier to get the range from a graph of the function (which is the topic
of a future section).
In this exercise, you are only asked to find the range for very simple functions.
EXAMPLES:
Question: What is the domain of the function f
defined by
f
(
x
)
=
x
+
2
?
Solution: Since any number inside a square root must be nonnegative,
we must have:
x
+
2
≥
0
(subtract 2 from both sides)
x
≥
-
2
Thus, using interval notation,
dom
(
f
)
=
[
-
2
,
∞
)
.
Question: What is the domain of the function f
defined by
f
(
x
)
=
3
?
Solution: This is a constant function! No matter what the input is, the output is the number 3.
For example,
f
(
0
)
=
3
,
f
(
-2.79
)
=
3
, and
f
(
π
)
=
3
.
All real numbers can be inputs.
Thus, using interval notation,
dom
(
f
)
=
(
-
&infty;
,
∞
)
.
Alternately, you could write
dom
(
f
)
=
ℝ
.
Question: What is the range of the function f
defined by
f
(
x
)
=
3
?
Solution: No matter what the input is, the output is the number 3.
Thus, the range contains only one number, 3.
Thus, using list notation,
ran
(
f
)
=
{
3
}
.
Question: What is the range of the function g
defined by
g
(
x
)
=
x
?
Solution: The outputs from square roots are always nonnegative.
Using interval notation,
ran
(
g
)
=
[
0
,
∞
)
.
Question: What is the domain of the function f
defined by
f(x)=-23x+5
?
Solution: The denominator cannot equal zero, so x cannot equal
-53 .
The "union" symbol, ∪, is used to "put sets together".
Thus,
dom(f)=
(-∞,-53)∪
(-53,∞)
.
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.