Power Functions: Case 2: a = 1/n - root functions

We continue our lessons on power functions. A power function is any function where the variable x is raised to a constant. That is...

Case 2. The power a = 1/n, where n is a positive integer. That is...

For instance, when n = 2, we have...

...the square-root of x. So these are what we term as root functions.

So examples of root functions are shown in the table below.

On the cartesian plane (i.e. the x-y plane), we can see that for even numbers of n, there exists only positive values of y. This is because we are unable to evaluate the square-root of a negative number as a real number. These are what we call "imaginary numbers" as they don't exist in the "real" world.

The domain and range of f(x) = x^1/n when n is even is [0, ∞.)

However, cubed-roots of negative numbers do exist in the "real" world. In fact, all real numbers have all odd roots. This is why for negative values of x, we also have negative values of y.

The graph of the square-root of x is really the top half of a horizontally oriented parabola x = y². All other graphs for even n are similar, except they become steeper and steeper for small values of x.

The graph of the cube-root of x is really y = x³ mirrored about the y-axis, then rotated 90° clockwise. With higher odd values of n, the graphs look similar but gradually get steeper around small values of x.

In these cases, the domain and range is (-∞,∞).

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All equations in this tutorial were created with QuickLatex

All graphs are created with www.desmos.com/calculator

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Here's a list of posts created so far on the topic of Functions:

1. Drawing piecewise defined functions
2. What are Polynomial Functions?
3. Power Functions: Case 1: a = n (n = 1,2,3...)
4. Power Functions: Case 2: a = 1/n

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