# Why is y=(-2)^x impossible to graph?

I have a question about why y=(-2)x is impossible to graph and I don't know what to say because I know that with some graph paper and a chart of values I can sure graph it pretty easily but my computer graphing program won't do it. Why not? Why is it "impossible" ?

What does it look like when you graph it on paper? Try doing it.

i did it and it's just a mirror image of 2^x

Mark44
Mentor
i did it and it's just a mirror image of 2^x
I don't think so.

I'm guessing that you graphed -2x, which is different from (-2)x. The graph of y = -2x is the reflection across the x-axis of the graph of y = 2x. The graph of y = 2-x is the reflection across the y-axis of the graph of y = 2x.

(-2)x has values that are not real for most values of x.

i did it and it's just a mirror image of 2^x

Are you sure? What is (-2)3/2?

Think about when x = 3/2, which would be y = (-2)3/2. This would be equivalent to y = √(-2)3. The square root of negative numbers involves imaginary numbers.

I really don't understand? (-2)^3/2 in undefined, but i can still do (-2)^2 and get 4?

Whenever $x$ is an integer, $(-2)^x$ will be a real number, and thus will be something you can plot on a graph. For any other value of $x$, it will involve taking the root of a negative number as Peppino said, which will result in an imaginary number that you can't plot on graph paper.

okay, i understand that now, but what is the significance of taking the root of a number in this situation?

Mark44
Mentor
okay, i understand that now, but what is the significance of taking the root of a number in this situation?

If x is a rational number such as 3/2, then you can write ax as a3/2. (Here I'm assuming that a > 0.) This is the same as (a3)1/2 or √(a3).

If the denominator of the rational number is even, then you're going to run into problems when the radicand is negative. That even denominator translates into a square root, fourth root, and so on.