# What if the exponent in a power function is negative?

Dec 18, 2014

TLDR: Long version:

If the exponent of a power function is negative, you have two possibilities:

• the exponent is even
• the exponent is odd

The exponent is even:

$f \left(x\right) = {x}^{- n}$ where $n$ is even.
Anything to the negative power, means the reciprocal of the power.
This becomes $f \left(x\right) = \frac{1}{x} ^ n$.
Now let's look at what happens to this function, when x is negative (left of the y-axis)
The denominator becomes positive, since you're multiplying a negative number by itself an even amount of time. The smaller$x$ is (more to the left), the higher the denominator will get. The higher the denominator gets, the smaller the result gets (since dividing by a big number gives you a small number i.e. $\frac{1}{1000}$).

So to the left, the function value will be very close to the x-axis (very small) and positive.
The closer the number is to $0$ (like -0.0001), the higher the function value will be. So the function increases (exponentially).

What happens at 0?

Well, let's fill it in in the function:

$\frac{1}{x} ^ n = \frac{1}{0} ^ n$
${0}^{n}$ is still $0$. You're dividing by zero! ERROR, ERROR, ERROR!!
In mathematics, it is not allowed to divide by zero. We declare that the function doesn't exist at 0.
$x = 0$ is an asymptote.

What happens when x is positive?

When $x$ is positive, $\frac{1}{x} ^ n$, stays positive, it will be an exact mirror image of the left side of the function. We say the function is even.

Putting it all together

Remember: we have established that the function is positive and increasing from the left side. That it doesn't exist when $x = 0$ and that the right side is a mirror image of the left side.

With these rules the function becomes:  