Root function as the inverse of the power function
The curve of y = x² allows
you to read off the values of the squares of numbers as well as
of their roots, because from y = x² follows x = 
. Given a =
4, you find from the curve that there belong to this ordinate the
values x = +2 and x = -2 - the values of 
. Similarly, 
yields +3 and -3.
The root function is the inverse of the power function. You simply interchange the dependent and independent
variables. Since
we always want to denote the independent variable by x,
we can interchange the variables in y = xn and
obtain 
.
Functions
which arise by interchange of x
and y
are called inverse
functions..
The graph of the inverse function is readily obtained from the original function.
Let
P1(a,b) and P2(b,a)
be two arbitrary, inverse points; then the triangles OP1Q1
and OP2Q2 are congruent, because they share
1) OQ1 =
OQ2 = b,
2) P1Q1 = P2Q2 = a,
3) 
P1Q1O
= 
OQ2P2 =
90º,
whence OP1 = OP2.
The triangle OP1P2 is therefore isosceles. The height, drawn from O to P1P2 , yields the axis of symmetry of this triangle. The points P1 and P2 lie therefore symmetrically with respect to OR. You can obtain one point from the other by reflection in OR.
OR is at the same time the bi-sector of the angle between the x-arrow and y-arrow, whence you have the Rule:
Inverse
points lie symmetrically with respect
to the coordinate system's bisector y = x,
whence :
The
image of the root function is the reflection
of the associated power function in the straight line y
= x.
The
image of y = 
is most simply obtained by reflection of y = x²
in the straight line y = x. You could have set up a
table of associated values of x and y and
entered the points on graph paper. A sufficient number of points
would have yielded for you the curve shown.
The graph tells you:
1) To each positive value of x
belong two y-values, which are equal and opposite
to each other. The function y = is therefore said to be multi-valued, its image is symmetric with respect to the abscissa.
2) There do not correspond y -
values to negative values of x. None of the numbers you
know so far are negative after they have been squared. The graph
of y = 
shows that every positive value of x corresponds to a
positive value of y, every negative value of x
to a negative value of y. In fact, the values
correspond to single values of y. The function y = is single-valued and centri-symmetric.
You can extend this investigation to arbitrary exponents. The reflected images you obtain have the name as the original curves.The last but one curve is a parabola of second order or just a parabola. The last curve is a parabola of third order. They differ by their locations. For higher order root functions
y = 
, etc.
you obtain curves similar to the two curves above, depending on whether the root exponent is even or odd.