This lecture will cover domain and range, notations for domain and range,
domain restrictions, domain of rational functions, domain of radical functions,
and additional restrictions on the domain of a function.
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The domain is the set of all x-values to which the function assigns a
corresponding real y-value. In other words, the domain is the set of x-values
where the function is defined. The range is the set of all y-values that result
from the x-values. Note that the domain and range are each sets of values.
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To represent the domain or range, there are three main notations. These
notations are Roster Notation, Set Builder Notation, and Interval Notation.
These three notations are best suited for different types of representations of
functions.
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The most straightforward notation for domain and range is roster notation.
Roster notation gives a list of all elements in a set. The list is enclosed by
braces, also known as curly brackets. Roster notation is most often used in
representing the domain and range of a set of ordered pairs, since there are
relatively few values to list in the domain and range. It is not advisable in
most cases to write the domain and range of an equation or a graph in roster
notation.
Example:
Since there are relatively few x-values for which this function is defined, use
the roster notation to list each value: Domain: {-8, 6, 0, -7, 10}
Similarly, list the range of this set of ordered pairs in roster form: Range: {0,
4, -3, -5}
Be sure to list each unique y-value in the range only once.
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A useful notation for describing a domain or range with a large number of
values is Set-Builder Notation. Set-Builder Notation gives a description of the
included values for the set. The form for Set-Builder notation is as follows:
the notation is enclosed in braces, like roster notation; after the opening
brace, the variable corresponding to the domain is given, followed by a
vertical bar; after the vertical bar, a description of the included values is
given. This description is sometimes referred to as the set of conditions for
inclusion into the set.
Set-Builder Notation is read in the following manner: “the set of all x (or y)
such that …” with the description of the included values following.
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The third notation is Interval Notation. In Interval Notation, a representation
of an interval on the real number line is given. Each interval is represented
by a lower bound value, followed by an upper bound value for the interval, in
ordered pair form. If a particular bound value is included in the set of values,
then that side of the ordered pair is marked by a square bracket instead of a
parenthesis. Because interval notation closely resembles the notation for
points on the graph, careful attention must be used when utilizing this form
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Shown here is the graph of a linear function. Note that the domain is the set
of all x-values where the function is defined. So the domain here represents
the set of all x-values for which there is a corresponding point on the graph,
which in this case for a line is all real numbers.
In set-builder notation, our description for the values of x is that x can be any
real number.
The range is the set of all y-values from each corresponding point on the
graph. Notice that for each real y-value, a corresponding point on the graph
can be located. So the range is also the set of all real numbers. The range
can be written in a similar manner as the domain.
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Shown here is the graph of a higher-order polynomial function. The domain
represents the set of all x-values for which there is a corresponding point on
the graph. For this polynomial it can be seen that the function is defined for
all real numbers.
In set-builder notation, the description for the values of x is that x can be any
real number.
The range is the set of all y-values from each corresponding point on the
graph. Notice that for each real y-value, a corresponding point on the graph
can be located. So the range is also the set of all real numbers.
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Shown here we have the graph of a linear function. The domain represents
the set of all x-values for which there is a corresponding point on the graph.
With this function the domain is defined for all real numbers.
In set-builder notation, the description for the values of x is that x can be any
real number.
Remember the range is the set of all y-values from each corresponding point
on the graph. Notice that in each corresponding point on the graph, the yvalue is -1. So the range is only -1.
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Shown here is the graph of a piecewise function. Remember that the domain
is the set of all x-values where the function is defined. So the domain
represents the set of all x-values for which there is a corresponding point on
the graph. So with this function the domain is defined between -5 and 4, but
not including -5 and -2.
In set-builder notation, our description for the values of x is {x | -5 < x <= 4
and x cannot equal -2}.
The range is the set of all y-values from each corresponding point on the
graph. Notice that for each real y-value, a corresponding point on the graph
can be located. So the range is every number between -3 and 3, but not
including -1.5 and 3). The range can be written in a similar manner as the
domain.
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Example:
This is an example of a function in equation form. Remember that f(x)
represents the function notation of this equation and can be rewritten as y =
x^2 -4.
The domain of this function is the set of all values of x for which this function
is defined.
For any real number x, the square of that number is defined as well as
taking the difference of this number and 4 to get back to a real number. So
the function is defined for any real number.
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Instead of trying to find where a function is defined, it is often much faster to
find where a function is NOT defined, or in other words, determine the
values that x cannot be.
So, begin by assuming that x can be any value and then find all x-values
where the function is undefined.
Some instances in equations where there are some restrictions on the value
of the variable include when the denominator of a fraction is zero, and when
the radicand of a radical is less than 0.
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A rational function is a quotient of two polynomial functions. Rational
functions are undefined for any values of x that cause the denominator to be
0. These values are restricted from the domain of the function.
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Example:
To see what values cause the denominator to be 0, set the denominator = 0
and solve for the variable. The resulting values will cause the denominator to
be 0 and should be excluded from the domain.
So setting the denominator equal to 0 in this example, results in
7x +28 = 0 -> 7x = -28 -> x = -4.
So at x=-4, the denominator is 0, and so f(x) is undefined. Any other realnumbered values are defined for this function.
In set-builder notation, the description of the domain should indicate that we
want to exclude this value: {x | x =/= -4 }.
In interval notation, remember that the domain is all real numbers, excluding
-4. So everything up to -4 and beyond -4, just not -4: (-inf, -4) U (-4, inf)
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Example:
Again start out by assuming that x can be any value, and then finding values
of x that cause the denominator to be equal to 0. To determine these values,
set the denominator of the function equal to 0 and then solve. The values
that cause the denominator to be zero can be found by factoring the left side
of this equation. So, if x = -2 or x =-3, the denominator of f(x) is zero and the
function is undefined at this point. These values will be restricted from the
domain.
In interval notation, the domain of this function would be: (-inf, -3) U (-3, -2)
U (-2, inf)
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A radical function is a function that contains the square root of a variable
expression. Rational functions are undefined for any values of x that cause
the radicand to be negative. These values are restricted from the domain of
the function.
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Example:
To see what values cause the radicand to be negative, set the denominator
>= 0 and solve for the variable. The resulting values will cause the redicand
to be positive and should be the domain.
So setting the radicand greater than or equal to 0 in this example, results in
x +7 >= 0 -> x >= -7.
So when x>=-7, the radicand is positive, and so f(x) is defined. Any other
real-numbered values are undefined for this function.
In set-builder notation, the description of the domain should indicate this
interval: {x | x >= -7 }.
In interval notation, remember that the domain is all real numbers greater
than or equal to -7. So everything beyond -7, including -7: [-7, inf)
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Example:
To see what values cause the radicand to be negative, set the denominator
>= 0 and solve for the variable. The resulting values will cause the redicand
to be positive and should be the domain.
So setting the radicand greater than or equal to 0 in this example, results in
-2x +7 >= 0 -> -2x >= -7 -> x <= 7/2 *Note that when multiplying or dividing
both sides of an inequality by a negative number reverses the inequality
symbol.
So when x<=7/2, the radicand is positive, and so f(x) is defined. Any other
real-numbered values are undefined for this function.
In set-builder notation, the description of the domain should indicate this
interval: {x | x <= 7/2 }.
In interval notation, remember that the domain is all real numbers less than
or equal to 7/2. So everything up to 7/2, including 7/2: (-inf, 7/2]
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Example:
Normally to see what values cause the radicand to be negative, set the
radicand >= 0 and solve for the variable. However, because this is a rational
function as well, the denominator cannot equal 0. So to see what values of x
fit into the domain, set the radicand strictly greater than 0. The resulting
values will cause the radicand to be positive and the denominator to not
equal 0.
So setting the denominator greater than 0 in this example, results in
5x - 45 > 0 -> 5x > 45 -> x > 9.
So when x>9, the radicand is positive and the denominator does not equal 0;
so f(x) is defined.
In set-builder notation, the description of the domain should indicate this
interval: {x | x > 9 }.
In interval notation, remember that the domain is all real numbers greater
than 9. So everything beyond 9, not including 9: (9, inf)
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For real-world behavior, there may be some necessary restrictions to the
domain.
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