In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers.
Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0. See Figure. We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers.
In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis to indicate that the endpoint is either not included or the interval is unbounded. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers.
Third, if there is an even root, consider excluding values that would make the radicand negative. See Figure for a summary of interval notation. First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.
Given a function written in equation form, find the domain. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers. Given a function written in an equation form that includes a fraction, find the domain. When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. Now, we will exclude 2 from the domain. Given a function written in equation form including an even root, find the domain.
When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Now, we will exclude any number greater than 7 from the domain.
Can there be functions in which the domain and range do not intersect at all? In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation.
Figure compares inequality notation, set-builder notation, and interval notation. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value.
If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is. Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included.
The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set.
For example,. Given a line graph, describe the set of values using interval notation. Describe the intervals of values shown in Figure using inequality notation, set-builder notation, and interval notation.
Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.
Given Figure , specify the graphed set in. Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x -axis. The range is the set of possible output values, which are shown on the y -axis.
Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. Figure Energy Information Administration. In interval notation, the domain is [, ], and the range is about [, ]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines. Given Figure , identify the domain and range using interval notation.
For example, the domain and range of the cube root function are both the set of all real numbers. We will now return to our set of toolkit functions to determine the domain and range of each. Both the domain and range are the set of all real numbers. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers. The same applies to the vertical extent of the graph, so the domain and range include all real numbers. Further, 1 divided by any value can never be 0, so the range also will not include 0.
Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative it is an odd function. Given the formula for a function, determine the domain and range. There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.
We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. We then find the range. What is the domain and range of a linear function? Is domain the independent or dependent variable? How do you find the domain and range of a function in interval notation?
How do you find domain and range of a rational function? How do you find domain and range of a quadratic function? How do you determine the domain and range of a function? What is Domain and Range of a Function? See all questions in Domain and Range of a Function. Well, let us get just a bit more complicated. Using interval notation we will show the set of number that includes all real numbers except 5. First, stated as inequalities this group looks like this:.
The statement using the inequalities above joined by the word or means that x is a number in the set we just described, and that you will find that number somewhere less than 5 or somewhere greater than 5 on the number line. In interval notation a logically equivalent statement does not use the word or, but rather a symbol for what is called the union of two groups of numbers. The symbol for union coincidentally looks like a U, the first letter of union.
However, it is really not a letter of the alphabet. Here is what the union symbol looks like:. So, the group of numbers that includes all values less than 5 and all values greater than 5, but does not include 5 itself, expressed as interval notation looks like this:. Let us consider one last set of numbers. We will consider a group of numbers containing all numbers less than or equal to 5 and also those numbers that are greater than 7 but less than or equal to Using inequalities this group of numbers could be notated like this:.
And using interval notation as described throughout this material this group would look like this:. We would interpret this interval notation as representing the total group of numbers as the union of two other groups. The first would start at negative infinity and proceed toward the right down the number line up to and including 5.
0コメント