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Tests

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Method

Horizontal Line Test

The horizontal line test is a graphical method to determine whether a given function is invertible. It involves drawing multiple horizontal lines over the entire range of a function. If any horizontal line intersects the function's graph more than once, the function is not invertible. For example, consider the following two functions.

Function I	Function II
$y = 0.1 x^{3} - x$	$y = - 0.05 x^{3} - 0.25 x$

To determine whether they are invertible, follow these two steps.

Draw the Function on the Coordinate Plane

Draw each function on the coordinate plane manually or using a graphing calculator or other mathematical software.

Draw a Horizontal Line and Look at the Intersection Points

Draw horizontal lines at different places through the coordinate plane. If one of the lines intersects the graph more than once, the function is not invertible. Conversely, if no horizontal line cuts the graph more than once, the function is invertible.

Three horizontal lines and their points of intersections with the functions I and II

Notice that

ℓ_{2}

cuts the first graph at three different points. This means that Function I does not pass the horizontal line test and, therefore, is not invertible. However, all of the horizontal lines drawn over Function II only intersect the graph one time at most. Because of this, Function II is invertible.

Function I is not invertible. $\times$
Function II is invertible. $✓$

Keep in mind that before stating whether a function is invertible, the drawn horizontal lines have to cover the entire range to ensure that no horizontal line cuts the graph more than once.

Why

Intuition Behind the Method

Recall that a function is called invertible if its inverse relation is also a function. For a function to be invertible, it has to be a one-on-one, or injective, function.

Injective Function
A function $f$ such that for every $x$ in the domain, there is exactly one unique $y$ that satisfies $f (x) = y .$

This is why drawing multiple horizontal lines (or one horizontal line, which then is moved across the entire range of the function) allows to check whether there are any

y -

values that belong to more than one

x -

value, making them not unique.

Note that during this test, it is assumed that the graph of a function continues without any significant change beyond the boundaries of the coordinate plane. If this were not the case, it could never be determined from a graph whether a function is invertible or not.

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