Inverse functions
Definition of inverse function
We shall introduce a new concept, that of inverse function,
in order to study further elementary functions. As was mentioned in the conceptof function, we are free to choose either x or y as independent variable in a functional
relationship between them, the actual decision being made purely as a matter of
convenience. Let us take a certain function y = f(x), with x as independent
variable.
The function defined by the same functional relationship y =
f(x), but having y as independent variable and x as the function
is called the inverse of the given function f(x), this
latter being often called the direct function. The notation for the variables
is not important in itself, so that, denoting the independent variable in both
cases by the letter x, we can say that
is the
inverse of f(x). For example, if the direct functions are
the inverse functions are
Finding the inverse function from the equation of the direct
function is called inversion.
Let us take the graph of the direct function y = f(x). It
can easily be seen that the same graph can serve as a graph of the inverse
function
In fact, both the equations
give the same functional relationship between x and y. Suppose
an arbitrary x is given in the direct function. If we mark off an interval from
the origin 0 along the axis X, corresponding to the number x, then erect a
perpendicular to X from the end of this interval as far as its intersection
with the graph, we obtain the value of y corresponding to the chosen x as the
length of this perpendicular, with the corresponding sign. In the case of the
inverse function
, we have only to measure off the given value y from the
origin 0 along the axis Y, a perpendicular to Y then being erected from the end
of this segment as far as its intersection with the graph.
The length of this perpendicular, with the relevant sign,
gives us the value of x corresponding to the chosen y.
The inconvenience arises here, that the independent variable
x is measured off on one axis, namely X, in the first case, whilst in the
second case the independent variable y is measured off on the other axis, Y. In
other words, we can only keep the same graph on transition from the direct
function y = f(x) to the inverse function 
Provided we bear in
mind that, on making the transition, the axis representing the value of the
independent variable becomes the axis representing the function, and vice
versa.
To avoid this inconvenience, we turn the plane as a whole on
making the transition, so that the axes X and Y change places. We simply turn
the plane of the figure, together with the graph, through 180° about the
bisector of the first quadrant of the coordinate axes. The axes having changed
places, the inverse function
now has to be written in the usual way
Thus, given the graph of the direct function y =f(x), the
graph of the inverse function 
is obtained simply by
turning the plane of the graph through 180° about the bisector of the first
quadrant of the coordinate axes.
The full-line curve of Fig. 25 represents a direct function,
the broken curve being its inverse. A broken line is also used for the bisector
of the first quadrant of the coordinate axes, the plane of the figure being
rotated about this, in order to obtain the broken curve from the full curve.
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