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UBC Calculus Online Course Notes

Powers of x: fractional exponents

We have already looked at the family of functions of the form $ y=x^n $ 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 sub-families, 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 $ y=x^{1/n} $ where n is a positive integer. These cousins are interesting not only in and of themselves, but also in relationship to the family $ y=x^n $ .

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 $ y=x^n= x \times x \times 
...x $ is already familiar as the function that multiplies x by itself n times while the function $ y=x^{1/n} $ extracts the n'th root of a number. The most familiar pair is

y & = & x^2 \\ 
y & = & x^{1/2} =\sqrt x 

Below, we illustrate the graphs of the two functions


where $ n $ is a positive integer. In our last discussion, we also included a multiplicative constant, $ a $ . We could do that here, too, but what we want to investigate shows up more clearly if we assume that constant is a=1.

  1. 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
    • for $ x \le 1  $
    • for $ x = 1  $
    • for $ x \ge 1  $

  2. Steepness
    • How steep are the graphs of $ y=x^{n} $ and $ y=x^{1/n} $ 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.

  3. 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 $ y=x^{n} $ and $ y=x^{1/n} $ ). 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.

  4. Challenge: On our previous page, we drew the functions $ y=x^{n} $ 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 $ y=x^{1/n} $ 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 $ y=x^{1/2} $ 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 $ y=x^{2} $ and $ y=x^{1/2} $ , for example, are inverses of one another. So are $ y=x^{3} $ and $ y=x^{1/3} $ , 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 $ y=(\sqrt{x})^2 $ 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.