How to Figure Out Domain & Range
The domain and range are aspects of a function which are often mentioned. The domain is a definition of all of the x-values of a function, and the range is a definition of all of the y-values of a function.
Most graphing calculators can find the domain and range of a function relatively easily. However, if you do not have access to a graphing calculator, these types of problems can also be solved by hand.
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Instructions
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1
Look at the function given. If it is in the form {(x1, y1), (x2, y2)...}, then the domain and range are simple to find: They are all of the x and y values of the function.
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2
List the numbers in the domain and range chronologically. For example, {(3,2), (-1,5) and (-2, -3)} would have a domain of {-2, -1, 3} and a range of {-3, 2, 5}.
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3
If the function given is an equation, find any instance in which a value substituted for "x" would result in a division of zero. For example, if a function of f(x) = 5/x, then x ≠ 0. Therefore, the domain is written as "all x ≠ 0".
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4
Set up a table with columns labeled "x" and "y". List seven points in the "x" column. Usually, -3, -2, -1, 0, 1, 2, and 3 will suffice. Substitute each value in the "x" column in the mathematical function and solve for y. For example, in the equation y=x^2-3, substituting "x" with "-3" results in y=(-3)^2-3, or 9-3. Therefore, y for "-3" would be "6".
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Take the points in the table you have created and graph them on graph paper with the horizontal axis being the "x" axis and the vertical axis being the "y" axis. You should see a shape formed by the points. Trace this shape with your pencil, making a line. If you do not see the shape, add more points to your table.
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Look at the graph and find any instance in which a value substituted for "y" would result in zero. An asymptote is usually indicative of this.
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Test the invalid "y" value by taking the "y" value found via this method and substitute it into the equation along with any "x" value in the domain as an ordered pair. If the value is invalid, then the test succeeds.
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