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).power function with positive integer exponent

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.power function with negative integer exponent

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)`.

power function with fractional exponent

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.