What if the exponent in a power function is negative?
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.
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:
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?
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!