
Powers of x: fractional exponents
We have already looked at the family of functions of the form
where n is a positive integer. Indeed, from our first discussion
of powers, we have already surmised that this "family" is actually
comprised of two subfamilies, the even and the odd powers, related
by certain common features, but distinct in others. We now extend our
investigation by looking at "cousins", also power functions, but
with fractional powers of the form
where n
is a positive integer. These cousins are interesting not only in and
of themselves, but also in relationship to the family
.
Ofcourse, before we get too deeply into the geometric and graphical
properties, we should remark that the functions are more commonly known
as powers and roots. The function
is already familiar as the
function that multiplies x by itself n times while the function
extracts the n'th root of a number. The most
familiar pair is
Below, we illustrate the graphs of the two functions
where
is a positive integer. In our last discussion,
we also included a multiplicative constant,
.
We could do that here, too, but what we want to investigate
shows up more clearly if we assume
that constant is a=1.
Questions:
 Small or large? Notice what happens to the
heights of the graphs as n increases.
In particular, notice that some parts of the graph are stretched
upwards while other parts shrink down. Comment on the behaviour
in the two graphs as n increases
 Steepness
 How steep are the graphs of
and
at the origin?
 How does the steepness of the graphs change as x increases?
Are they the same or different?
 How is the steepness of the graph affected by the
value of n? If the answer is different at various portions of the
curves, indicate where the slope gets shallower or steeper as n
increases.
 By experimenting with the above graphs for various n values,
explain how the power function and its fractional power "cousin"
are related (i.e explain the relationship between the graphs of
and
). This should be done in the
context of the line y=x and it should involve the concept of
symmetry once more. The word "reflection"
will be useful in this discussion.
 Challenge: On our previous page, we drew the functions
for both positive and negative values of x.
(In fact, this led to the discussion of even and odd functions and
symmetry properties.) But here, we compared the graphs to those
of
for positive x values only. Why ? What
might go wrong if we included the negative x axis ?
(Hint It may help to you remember what one typical example,
say
actually "means". Then think of what happens
when you try to "plug in" negative x values.

A word about inverse functions
What you have discovered is a general property characteristic
of pairs of inverse functions. The functions
and
, for example, are
inverses of one another. So are
and
, and so on. We will discuss this in much
more detail later, but essentially this just means that one function
reverses the operation of the other. The first example,
the function that squares a number seems quite
clearly the "inverse" of the
function that takes square roots.
But here, we should be a little careful.
A thorough discussion of inverse functions requires
attention to detail, and is thus postponed until we are ready to
tackle some technical matters. To illustrate the kinds of problems
that might come up, our simple introductory example will do well.
Suppose we take the square root of a
a number, and then square the result. We "expect" to get the original
number back again. (This is usually the case, no problem  but
we have to restrict ourselves to positive numbers or the
first step won't work!)
Is this also true if we reverse the order of the
operations? First the square, then the square root? What happens if we
start with a negative number, like x=4 and perform the two operations?
Do we get the original number back? Evidently not!, in fact, the
combined operation
behaves just like the
absolute value function! Only if we restrict attention to positive x
values are the functions really inverses of one another. We will
discuss this again later, in greater detail.

