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Unformatted text preview: The Horizontal Line Test: A function f is one-to-one if and only if no
horizontal line intersects the graph of f more than once.
We say that the graph of a function passes the Horizontal Line Test if no horizontal line intersects
the graph more than once; otherwise, we say the graph of the function fails the Horizontal Line
Test. We have argued that if f is invertible, then f must be one-to-one, otherwise the graph given
by reﬂecting the graph of y = f (x) about the line y = x will fail the Vertical Line Test. It turns
out that being one-to-one is also enough to guarantee invertibility. To see this, we think of f as
the set of ordered pairs which constitute its graph. If switching the x- and y -coordinates of the
points results in a function, then f is invertible and we have found f −1 . This is precisely what the
Horizontal Line Test does for us: it checks to see whether or not a set of points describes x as a
function of y . We summarize these results below. 5.2 Inverse Functions 297 Th...
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