Power functions > Power functions
12345Power functions

## Theory

Take a look at the applet: Power Functions

Here you see the graphs of the power function $f\left(x\right)={x}^{p}$ at different values for $p$. The function has the following properties for $x>0$:

• $p>1$: the curve goes through points $\left(0,0\right)$ and $\left(1,1\right)$ and has an increasing (positive) slope.

• $p=1$: $f$ is a linear function through points $\left(0,0\right)$ en $\left(1,1\right)$.

• $0: the graph goes through points $\left(0,0\right)$ en $\left(1,1\right)$ and has a decreasing (positive) slope.

• $p<0$: the function is undefined for $x=0$, the graph goes through point $\left(1,1\right)$ and has a decreasing (negative) slope, the $x$ -axis and the $y$ -axis are asymptotes of the graph.

For $x<0$ the function only exists if $p$ is a whole number is (or if $p$ is a fraction with an uneven denominator, such as $\frac{1}{3}$, $\frac{2}{3}$, $\frac{1}{5}$, $\frac{2}{5}$, etc). Depending on whether $p$ is positive or negative, the graph will be increasing or decreasing.

The equation ${x}^{p}=a$ has exactly one solution when $a>0$, and as long as $p$ is not an even whole number (not $0$), because in that case there would be two solutions. The equation ${x}^{p}=a$ has exactly one solution when $a<0$ and if $p$ is an uneven whole number (not $0$).