Understanding the Horizontal Line Test | Determining If a Function is One-to-One or Not

horizontal line test

The horizontal line test is a method used to determine if a function is one-to-one or injective

The horizontal line test is a method used to determine if a function is one-to-one or injective. It is called the horizontal line test because it involves drawing horizontal lines on a graph of a function.

To apply the horizontal line test, you need to consider the graph of the function. If every horizontal line intersects the graph of the function at most once, then the function is one-to-one. In other words, no two different input values of the function produce the same output value.

If there exists any horizontal line that intersects the graph of the function at more than one point, then the function fails the horizontal line test, and it is not one-to-one.

For example, consider the function f(x) = x^2. If we draw a horizontal line across the graph of this function, it will intersect the graph at two points for any y-value greater than or equal to 0. This means that the function fails the horizontal line test and is not one-to-one.

On the other hand, consider the function g(x) = 2x + 3. If we draw a horizontal line across the graph of this function, it will intersect the graph at most once, no matter the y-value. Therefore, the function passes the horizontal line test and is one-to-one.

The horizontal line test is particularly useful in determining whether a function has an inverse, as one-to-one functions have unique inverses. If a function fails the horizontal line test, it does not have an inverse.

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