# Power Function

Function of the form f(x)=x^n where n is constant is called power function.

Depending on value of n graph of power function has different forms and properties.

Case 1: n is positive integer.

When n=1 we obtain line y=x. When n=2 we obtain parabola y=x^2. When n=3 we obtain cubic function y=x^3.

Graph of function depends on whether n is even or odd.

If n is even then functions of the form f(x)=x^n are even and their graphs are similar to the graph of function y=x^2.

If n is odd then functions of the form f(x)=x^n are odd and their graphs are similar to the graph of function y=x^3.

Note, that when |x|<1 the bigger n, the closer graph to x-axis. For |x|>1 the bigger n, the faster functions grows (the further from x-axis). Case 2: n is negative integer, i.e. n=-a where a is positive integer.

Domain of such functions is all x except x=0 (function is nor defined when denominator equals 0).

When n=1 we obtain hyperbola y=1/x.

Graph of function depends on whether n is even or odd.

If n is even then functions are even and their graphs are similar to the graph of function y=1/x^2.

If n is odd then functions are odd and their graphs are similar to the graph of function y=1/x.

Note, that when |x|<1 the bigger n, the furthe graph to x-axis. For |x|>1 the bigger n, the closer function to the x-axis. Case 3: n is irreducible fraction, i.e. n=a/b where a and b are integers.

Domain of such functions depends on a and b. Remember that we can't extract even-degree root of negative number.

First consider case when a/b is positive.

In this case x^(a/b)=root(b)(x^a). If b is odd then domain is interval (-oo,oo). If b is even then a is odd (remember that a/b is irreducible, so a and b can't be both even) then domain is interval [0,+oo).

Now consider case when a/b is negative, i.e. a/b=-m/k where m and k are positive integers.

In this case x^(a/b)=1/(root(k)(x^m)). If k is odd then domain is interval (-oo,oo), except x=0. If k is even then m is odd (remember that m/k is irreducible, so m and k can't be both even) then domain is interval (0,+oo). For example, domain of y=x^(5/2) is (0+oo). Domain of the y=x^(-5/2) is (0+oo).

For case when n is rational graph will lie between graphs of power functions with closest integer values.

For example, graph of the function y=x^(5/2) will lie between graphs of functions y=x^2 and y=x^3 because 2<5/2<3. See figure to the left.