Inverse Function: Graph

y = f(x) and y = f^-1(x)
show the reflection in y = x.

y = f(x) is x = f^-1(y).

Inverse Function

See x = f^-1(y) and y = f^-1(x).
x and y are switched.

So y = f(x) and y = f^-1(x)
show the reflection in y = x.

So doing the horizontal line test
for y = f(x) is

doing the vertical line test
for its inverse y = f^-1(x).

So, if y = f(x) passes the horizontal line test,
then y = f^-1(x) passes the vertical line test.
Then y = f^-1(x) exists.

This means
if y = f(x) passes the horizontal line test,
y = f(x) has an inverse function y = f^-1(x).

To see if y = f(x) has an inverse function,
do the horizontal line test.

From the bottom side of the graph,
think of a horizontal line.
Move the horizontal line to the top side.
And see if the horizontal line and the graph
intersect at [one point].

As you can see,
the line and the graph intersect at one point.

So y = f(x) passes the horizontal line test.

So y = f(x) has an inverse function.

To see if y = f(x) has an inverse function,
do the horizontal line test.

If the horizontal line is like this,
the line and the graph intersect at one point.

So, for this case,
y = f(x) passes the test.

But if the horizontal line is like this,
the line and the graph intersect at two points:
not at one point.

So y = f(x) fails the horizontal line test.

So y = f(x) does not have an inverse function.