What is the horizontal line test?

In January of this year, we have discussed about the vertical line test. We have learned that if a vertical line intersect a graph more than once, then that graph is not a function. In this post, we learn about the horizontal line test and its relation to inverse functions.

Suppose we have a function f. Then, we input x and call the output f(x). If we do things backward, suppose we have the number y which is in the range of f. What value should we input to f to get y? Let’s have a more specific example.

Suppose we have the function f(x) = x^2. And we choose the number y = 9 which is in the range of f. What number should we input in f to get 9? Well, we will have two numbers, those are 3 and -3.

On the other hand, suppose we have the function g(x) = x^3. And we choose the number y = 8. What number should we input in g to get 8? We only have one and only one number which is equal to 2. Now this is true for any y in g since there is only one cube root of any number.

Now if there is one and only one value of x which satisfies f(x) = y for each y, then we can define a new function which reverses the transformation. Given the output y, there is one and only one input x that results to y. This function is called the inverse function.

Notice from the graph of f below the representation of the values of x. Graphically, y = 9 is a horizontal line, and the inputs x = 3 and x = -3 are the values x at the intersection of the graph and the horizontal line. In fact, if you put a horizontal line at any part of the graph except at y = 0, there are always 2 intersections. As we have learned in functions, for every input, there is exactly one output and hence reversing f(x) = x^2 will have two outputs. Thus, it cannot qualify as a function and therefore it has no inverse function.

horizontal line test 1

However, if we examine the graph of g(x) = x^3, a horizontal line can only intersect it at one point regardless of where you place it. Therefore, g has an inverse function.

horizontal line test 2

From the discussion above, we can conclude that we can use horizontal lines to test whether a function has an inverse or none. That is, if every horizontal line intersects the function at most once, then the function has an inverse; otherwise, it has no inverse function. This method is called the horizontal line test.

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