Chapter 1 | Functions and Graphs
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input to exactly one output. For example, let’s try to find the inverse function for f ( x ) = x 2 . Solving the equation y = x 2 for x , we arrive at the equation x = ± y . This equation does not describe x as a function of y because there are two solutions to this equation for every y >0. The problem with trying to find an inverse function for f ( x ) = x 2 is that two inputs are sent to the same output for each output y >0. The function f ( x ) = x 3 +4 discussed earlier did not have this problem. For that function, each input was sent to a different output. A function that sends each input to a different output is called a one-to-one function.
Definition We say a f is a one-to-one function if f ( x 1 ) ≠ f ( x 2 ) when x 1 ≠ x 2 .
One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the xy -plane, according to the horizontal line test , it cannot intersect the graph more than once. We note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one ( Figure 1.38 ).
Rule: Horizontal Line Test A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.
Figure 1.38 (a) The function f ( x ) = x 2 is not one-to-one because it fails the horizontal line test. (b) The function f ( x ) = x 3 is one-to-one because it passes the horizontal line test.
Example 1.28 Determining Whether a Function Is One-to-One
For each of the following functions, use the horizontal line test to determine whether it is one-to-one.
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