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Appendix C C-67
PROJECT
Inverse Secant Functions
It is interesting to note that the property of periodicity, fundamental to the importance
and usefulness of the sine, cosine, and tangent functions, is precisely
what prevents these functions from being one-to-one and therefore having inverse
functions. So in Section 9.5* we restricted the domains of these functions
to obtain one-to-one versions and proceeded to construct the inverse sine, inverse
cosine, and inverse tangent functions, systematically following our development
of inverse functions from Chapter 3. The domains we chose are the
ones that are traditionally used for this purpose.
In the case of the secant function there are two common choices of domain.
These result in two one-to-one versions of the secant function, each with its
own distinct inverse secant function. Before we start, keep in mind that our
choice of domain is made to satisfy two conditions: We need a version of the
function that is both one-to-one and has the same range as the unrestricted secant
function.
The secant function is not one-to-one. Why? Looking at the graph of
y sec x where the secant’s range is obtained, we can restrict the domain to the
union of intervals 30, p 2 2 3p, 3p 2 2 to obtain a version of secant with the correct
range. We have our first restricted secant function:
y sec x
with
domain 30, p 2 2 3p, 3p 2
2 and range (q, 1] [1, q)
which is one-to-one. Its graph is shown in Figure A(i).
y
y
3π
2
(π, -1)
-1
π
2
π
(π, -1)
3π
2
x
π
(-1, π)
-1
π
2
(1, 0)
y=sec -1 x
x
(i) The graph of y=sec x for x in
π 3π
[ 0, ) ∪ [ π,
2 2 )
Figure A
(ii) The graph of y=sec -1 x
*Precalculus: A Problems-Oriented Approach, 7th edition