2.4 Inverse functions 39
In physical applications it is usually obvious from the context which value is to be
chosen. It is also seen, when xand yare real numbers, that whereas yis defined for all
values ofx,−∞ 1 < 1 x 1 < 1 +∞,xis only defined fory 1 ≥ 11.
0 Exercises 30, 31
The finding of the inverse function is not always so straightforward.
EXAMPLE 2.11y 1 = 1 f(x) 1 = 1 x
51 − 12 x
In this case the inverse functionf
− 1exists for all values of x, but it cannot be written
in simple algebraic form, although it can be tabulated and plotted as in Figure 2.5.
As in Example 2.10, graph (b) has been obtained from graph (a) by rotation around
the linex 1 = 1 y.
Figure 2.5
In Example 2.11 the functional dependence of yon xis given explicitly by the
right side of the equation; yis an explicit functionof x. On the other hand, xcannot
be written as an explicit function of y, and the equation defines xas an implicit
functionof y. In general, an equation of the form
f(x,y) 1 = 10
wheref is a function of both xand y, gives either variable as a function of the
other. In many equations in the physical sciences it is either not possible to express
one variable as an explicit function of the others, or it may not be convenient to
do so.
........
...
..
...
...
..
...
...
..
...
...
..
.................
........................− 1. 5 1. 5
− 2
2
................................................................................................................................................................................................................y
x
x=y
...........
...........
...........
............
...........
..........
...........
............
...........
...........
............
...........
............
...........
..........
.............
..........
...........
............
...........
...........
............
...........
...........
.............
...........
...........
..............
...........
...........
.............
.............
.............
.................
...............................................
...............
...........
...........
...........
..........
...........
.............
...........
..........
...........
.........
..........
..........
.........
.........
..........
..........
.........
............
..........
...........
.............
...........
............
............
..........
...........
............
..........
..........
...........
..........
.........
...........
..........
.........
...........
.........
.........
..........
..........
.........
..........
.........
.........
..........
.........
.........
..........
.........
..........
..........
.........
.........
..........
..........
..........
..........
.........
...........
..........
.........
............
..........
..........
............
...........
...........
...........
..........
..........
..........
...........
.........
..........
..........
..........
.........
...........
.........
..........
..........
.........
...........
.........
..........
..........
..........
..........
...........
..........
..........
............
..
...........
............
...........
..........
.............
..........
...........
............
...........
...........
............
...........
..........
.............
..........
...........
............
...........
...........
............
...........
...........
.............
...........
...........
............
............
...........
.............
............
...........
..............
..............
...............
...........................................
..............
...........
............
..........
..........
............
............
...........
...........
..........
..........
..........
.........
.........
...........
.........
.........
...........
..........
..........
............
............
...........
.............
...........
..........
............
...........
..........
...........
..........
..........
...........
.........
..........
...........
.........
.........
...........
.........
.........
..........
.........
.........
...........
.........
.........
..........
.........
.........
..........
.........
.........
...........
.........
.........
...........
.........
.........
...........
..........
..........
...........
..........
..........
............
...........
...........
...........
..........
..........
...........
..........
..........
..........
..........
.........
...........
.........
.........
...........
.........
.........
...........
.........
..........
...........
.........
..........
...........
..........
..........
............
...
(a)y=f(x)
...............
.................................
...
..
...
..
...
...
...
..
...
..
...
.− 1. 5
- 5
− 2 2
................................................................................................................................................................................................................x
y
x=y
....
.....................
......................
........................
.....................
......................
........................
......................
.....................
........................
....................
.......................
....................
....................
....................
.....................
.................
.................
.............
.............
............
............
..........
...........
.............
................
......................
..........................
..................................
...........................................
..............................................
....................................................................
.......................................................................................
...........................................................................
..............................................................................
......................................................................................................
.......................................................................................................................................
......................................................................................................................................
..................................................................................
...................
........................
......................
.....................
.........................
.....................
.....................
.........................
.....................
.....................
.......................
....................
....................
......................
...................
.................
..................
..............
.............
..............
...........
..........
...........
............
...............
....................
........................
................................
..........................................
...........................................
.....................................................................
...............................................................................
...........................................................................
.............................................................................
................................................................................................
...............................................................................................................................
....................................................................................................................................................
........................................................................................................
(b)x=f
− 1(y)