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:

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

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.

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

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:

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 the 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.