Horizontal Line Test

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In this lesson, we will delve into the concept of one-to-one functions and explore a method called the horizontal line test to determine if a function is one-to-one. As we already know, a function is a special type of relation where each x-value corresponds to one and only one y-value. In the case of one-to-one functions, this relationship extends further, ensuring that for each y-value, there is only one x-value. Thus, one-to-one functions exhibit a unique and distinct mapping between x and y values. The Horizontal Line Test: To determine if a function is one-to-one, we can employ the horizontal line test. This test involves using horizontal lines of the form y = k, where k represents a constant value. By examining the points of intersection between the graph of the function and the horizontal line, we can make conclusions about the function's one-to-one nature. For a given function, consider a horizontal line with a fixed value of y (k). As we move this line from left to right across the graph, we observe the points of intersection. If the horizontal line intersects the graph in more than one location, it implies that the same y-value (k) is associated with multiple x-values. In such cases, the function fails the horizontal line test and is not one-to-one. On the other hand, if the horizontal line intersects the graph at most once, with no repeated intersections, it indicates that each y-value (k) corresponds to a unique x-value. This scenario satisfies the criteria for a one-to-one function.
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