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

• Gregory.gags
In summary: However, if the denominator of the rational number is odd, then the imaginary number will cancel out the imaginary part of the radicand, leaving only the real part. This is why (-2)^3/2 is undefined but (-2)^2 is real.

#### Gregory.gags

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

Gregory.gags said:
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.

Gregory.gags said:
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?

Gregory.gags said:
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.

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

The equation y=(-2)^x is impossible to graph because it represents an exponential function with a negative base. In exponential functions, the base must be a positive number in order to have a well-defined graph. Negative bases result in non-real solutions, making the graph impossible to plot on a traditional Cartesian plane.

## 2. Can't we just use a calculator or computer to graph y=(-2)^x?

While some advanced graphing calculators or computer programs may be able to plot complex functions, they still rely on the basic principles of graphing. In this case, the negative base of (-2)^x would still result in non-real solutions, making it impossible to graph.

## 3. What if we change the equation to y=(-2)^(-x)?

Even with a change in the exponent, the negative base of (-2) would still result in non-real solutions and an impossible graph. In order to graph an exponential function, the base must be a positive number.

## 4. Are there any other ways to graph y=(-2)^x?

No, there are no other ways to graph y=(-2)^x. As previously mentioned, exponential functions with negative bases cannot be graphed on a traditional Cartesian plane. Other methods of graphing, such as using a polar coordinate system, would also not be able to accurately represent this function.

## 5. Can we still study the behavior of y=(-2)^x without graphing it?

Yes, we can still study the behavior of y=(-2)^x without graphing it. We can analyze the equation algebraically and look at the properties of exponential functions with negative bases. Additionally, we can use technology to plot the function for a limited range of values, providing us with a visual representation of its behavior.