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Is the Inverse a Function? (page 2 of 7)

Sections: Definition / Inverting a graph, Is the inverse a function?, Finding inverses, Proving inverses

Now suppose your function is { (1, 2), (2, 1), (3, 4), (5, 1) }. The inverse of this function is { (2, 1), (1, 2), (4, 3), (1, 5) }. This inverse has two points, (1, 2) and (1, 5), that share a common x-value but have different y-values. This means that the inverse is NOT a function.

  

Graphically, the original function looks like this:

  

Draw the dots
  

  

You can find the inverse algebraically, by flipping the x- and y-coordinates, or graphically, by drawing the line y = x...

  

Draw the reflection line and the lines from the points across the reflection line
  

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...and reflecting all the points across it:

  

Draw in the new (reflected) dots
   

Note that it's perfectly okay for the inverse to "overwrite" the original function's points! The points "(2, 1) and (1, 2)" of the inverse overwrote the points "(1, 2) and (2, 1)" of the original function, which is why the graph is "missing" a red dot.

  

The inverse is marked in blue
   

  

But you can see that the inverse is not a function: there are two points sharing an x-value.

  

Vertical Line Test: the inverse is not a function
  

There is a quick way to tell, before going to the trouble of finding the inverse, whether the inverse will also be a function. You've seen that you sort of "flip" the original function over the line y = x to get the inverse.
  

  

Using this fact, someone noticed that you can also "flip over" the Vertical Line Test to get the Horizontal Line Test. As you can see, you can draw a horizontal line through two of the points in the original function:

  

Horizontal Line Test: the inverse will not be a function

Since the original function had two points that shared the same Y-VALUE, then the inverse of the original function will not be a function. This means, for instance, that no parabola (quadratic function) will have an inverse that is also a function.

In general, if the graph does not pass the Horizontal Line Test, then graphed function's inverse will not itself be a function; if the list of points contains two or more points having the same y-coordinate, then the listing of points for the inverse will not be a function. So when you're asked "Will the inverse be a function?", if you're given a graph, draw a horizontal line; if you're given a list of points, compare the y-coordinates.

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Cite this article as:

Stapel, Elizabeth. "Is the Inverse a Function?." Purplemath. Available from
    http://www.purplemath.com/modules/invrsfcn2.htm. Accessed
 

 

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