Now that we have a meaning for the expression
x1/n, we can consider the function f(x)=x1/n.
For a given value of x, we can work out f(x). Then joining all such pairs (x,f(x)) we obtain the graph of the function. Before we plot the graphs, we will consider some features of the function.
Firstly, are there any values of x such that f(x) is not defined (as
with our function "x to the -1" at x=0)?
Well, any positive
number has a square root. The square root may not be a whole number (and
it usually isn't). Similarly every positive number has a cube root and a
4th root and so on.
But what about negative values of x? There is
no number which when multiplied by itself gives -16. So -16 has no
square root. Indeed no negative number has a square root. What about cube
roots?
Well, if you multiply a negative number by itself three times, the result is negative.
So negative numbers do have cube roots.
To sum up then:
if n is even then f(x) is not defined for negative values of
x,
if n is odd then f(x) exists for
all values of x.
Secondly, is there only one value of f(x) for each value
of x?
The functions we have looked at so far have been
single-valued, this means that for each value of x, if there
is a value of f(x) then there is only one of them.
But if
x=16, say, then there are two values of f(x), because 4x4=16 and
(-4)x(-4)=16. We say therefore that "x to the power a half" is a
multi-valued function.
By convention if we write "x to
the power a half" we mean the positive square root.
Every positive number has two square roots, one is positive and
the other negative. What about cube roots? If a positive number is cubed
the result is positive, if a negative number is cubed the result is
negative. So the cube root of a positive number must be positive and it
has only one cube root. Similarly every negative number has only one cube
root, so the function "x to the power a third" is
single-valued.
Now consider what happens to "x to the power a half" (i.e. the square root of x) as x gets small. Below is a table of x and the square root of x for small values of x.
As x gets small, the square root of x gets small too, but more
slowly than x.
If x is less than one, then the square root of x is
bigger than x.
If x is greater than one, then the square root of x is
smaller than x.
Now what happens if x gets large?
Below is a table of large values of x and the
corresponding values of the square root of x.
As x gets larger, the square root of x gets larger too, but more
slowly.
Now look at the graph of the function: use the plotter to plot the function x**.5 (which is x to the power half).
Note that the plot is of the positive square root only. The curve of the negative square root is a reflection of this curve in the x-axis. You can see this by inserting a - sign before the x**.5 in the form. Try this now.
Now use the plotter to see a cube root curve. (So you need to put in x**1/3).
Try altering the power now to see a plot of the
function "x to the power a quarter". Then experiment with
different values.
You will see that the curves are all similar,
but that as n gets larger, the curve gets steeper near the origin, and
shallower for large x.
All the fractional powers in the graphs above have been positive.
The differences between the positive-fractional-power functions and the
negative-fractional-power functions mirror the differences between
positive-integer-power functions and negative-integer-power functions,
discussed in the section on negative powers.
Choose a value for n and plot the function x**-1/n, a
negative-fractional-power function.
For example you might try plotting
the function x-1/2.
You should see that the curves of the negative fractional power functions are similar, in that the functions tend to infinity as x tends to zero, and the functions tend to zero as x tends to infinity.
Finally, you may think we haven't covered all fractional powers
yet, only those of the form x1/n,
where n is a whole number,
so what about "x to the power
two-thirds" for instance?
The answer is that by using the
third power law we can see that
x2/3 = (x1/3)2,
which is the cube root of x, squared.
In this way, all fractional
powers can be reduced to some power of a function of the form
x1/n.