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In each of these relations, our source set is the entire set of real numbers, and that s not necessarily the domain Also, our destination set is the entire set of reals, and that s not necessarily the range: In Example 1, we subtract 1 from each value of the independent variable to get a value of the dependent variable This operation produces a one-to-one correspondence between the set of real numbers and itself For every value we input, we get a unique output Also, every output value is the result of one and only one input value It follows that this relation is a bijection
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Figure 11-1 The domain of a relation is a
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subset of the maximal domain The range is a subset of the co-domain
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In Example 2, we square each value of the independent variable to get a value of the dependent variable For every value of the dependent variable except 0, two different values of the independent variable are assigned to it The relation is not an injection, because it s not one-to-one It can t be a bijection, then, either The independent variable can attain any real value, but the dependent variable can never be negative, so this relation is not a surjection onto the set of real numbers We must therefore classify this relation as none of them In Example 3, we take the positive or negative square root of each value of the independent variable to get a value of the dependent variable The independent variable can t be negative, but the dependent variable can be any real number The relation is therefore a surjection onto the set of real numbers But it s not an injection, because most values of the independent variable map to two values of the dependent variable It s not a bijection then, either In Example 4, we take the nonnegative square root of each value of the independent variable to get a value of the dependent variable As in Example 3, the independent variable can never be negative Neither can the dependent variable In this case we don t have an injection, because some real numbers in the source set don t have any counterparts in the destination set We don t have a surjection either, because the range fails to cover the entire set of real numbers We must categorize this as a none of them relation
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What s a Two-Space Function
In two-space, a function is a relation that never maps any value of the independent variable to more than one value of the dependent variable All functions are relations, but not all relations are functions Figure 11-2 shows Venn diagrams of a legal assignment for a function (left) and an illegal assignment (right)
The vertical-line test In the Cartesian xy plane, suppose that x is the independent variable, and we plot it against the horizontal axis Also suppose that y is the dependent variable, and we plot it against the vertical axis When we see the graph of a simple relation, it usually appears as a line or curve More complicated relations may graph as groups of lines and/or curves We can test the graph of any relation in the Cartesian xy plane to see if it represents a function of x Imagine an infinitely long, movable vertical line that s always parallel to the dependent-variable axis (the y axis) Suppose that we re free to move the line to the left or right, so it intersects the independent-variable axis (the x axis) wherever we want If the graph is a function of x, then the movable vertical line never intersects the graph of our relation at more than one point If, in any position, the vertical line intersects the graph at more than one point, then the relation is not a function of x We call this exercise the vertical-line test
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