Power Functions: Case 3: a = -n: reciprocal functions

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

Case 3. The power a = -n, where n is a positive integer. That is...

When n = 1, we have...

...which is the simplest form of reciprocal functions. You may have previously seen that when graphing this function, the result is a hyperbola as shown below...

The curves of hyperbolas are always split into 2 branches. In the case of f(x) = 1/x, the coordinate axes serve as the branches' asymptotes.

Reciprocal functions have some interesting properties:

At x = 0, the value of the function is indeterminate or undefined.

We can see that as x approaches 0 from left or the right, the magnitude of f(x) becomes infinitely large, but not only that, is it an infinitely large negative number or positive number? Actually, this question is non-sensible as infinity can only be a concept, with no defined value.

For infinitely large magnitudes of x, the curve is assumed to meet the x-axis, but since there is no value for ∞, we can not be sure.

Thus we can only say the function approaches zero for increasing magnitudes of x.

There are no local minimum or maximum values.
They are infinitely differentiable except at x = 0
They are infinitely integrable except at x = 0

The domain of f(x) = 1/x is...

...or in plain English, all values of x except for x = 0.

Similarly, the range of the function is...

All equations in this tutorial were created with QuickLatex

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

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

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

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