Calculus: Analytic Geometry and Calculus, with Vectors

8.3 Inverse trigonometric functions 459

in D1 for which f(x) = y. Then we can introducea function fl, which

is different from f because it has a different domain, for whichfi(x) = f(x)

when x is in Di and for which fi(x) is undefined whenx is not in D1. This

function f, is called the restriction of f to D1. Wenow have a general

situation that may be easier to visualize than its applicationsto trigono-

metric functions. The schematic Figure 8.31 shows the domain and

Domain Range

D,off1 f Roffi

Y

Range Domain

of fi ' fl of fi l

(^0) D1 x x+Ax x
Figure 8.31 Figure 8.32
range of fi, and the upper arc from x to y can make us think of a path
along which fi might carry x to the y which is fi(x). The function fi
sets up a one-to-one (or schlicht) correspondence between D1 and R,
that is, to each x in D1 corresponds exactly one y in R for which f(x) = y
and to each y in R corresponds exactly one x in D1 for whichAx) = y.
The function with domain R and range Di which carries eachy in R
into the x in D1 for which fi(x) = y is called the inverse off, and is denoted
by As indicated by the figure, A-1 undoes what f, does. If fi(x) = y,
then x Moreover, fi1 (f1(x)) = x when x is in D and
MANY)) = y
when y is in R.
Figure 8.32 shows the graph of a particular function f to which the
following theorem applies.
Theorem 8.33 If f is continuous over an interval D1 in El and if f has a
derivative for which f'(x) > 0 at each inner point x of D1 [or f'(x) < 0 at
each inner point x of D1] then the restriction fi of f to D1 has a range R
which is an interval and has also an inverse f,-' which is differentiable at
each inner point y of R. Moreover, the first of the formulas
(8.331) A _"(Y)

• f'1(fil(Y))'

dx 1

dY - dy

dx

is valid when y is an inner point of R.
It seems to be possible to put the first formula of (8.331) in the second
form without sacrifice of meaning. To prove the theorem, we use the
hypotheses to conclude that fi is increasing (or decreasing) over Di and
that (because of the intermediate-value theorem) R is an interval. Hence

Y+Iy

Y