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

##### 1 Answer

**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:**

Anything to the negative power, means the reciprocal of the power.

This becomes

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

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

**What happens at 0?**

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

**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.

**What happens when x is positive?**

When

**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

With these rules the function becomes:

**What about an odd exponent?**

The only change with an odd exponent, is that the left half becomes negative. It is mirrored horizontally. This function becomes:

Hope this helped!