Expert: Josh - 7/13/2004

Question
Is y=-3^x a exponential function? The reason I say it might, is:the equation for an exponential function is y=ab^x, so maybe a is a negative 1, and the b is 3.Thus, it's a exponential function because b is greater than 0 and not equal to one. Also, if I'm correct, this raises another question, why does a textbook have to say b>0,, when will it ever be? You can always say a is negative.
Thank you for your time and am looking forward to hearing back from you.

Answer
Ab,

To answer your question, we have to pay respect to the associative law in mathematics.

I presume that you original equation reads y=-(3^x).
In this case, yes, we have a negative exponential function.
Since 3^x must be non-negative, it approaches 0 asymptotically as x-> -infinity, it crosses y=1 when x=0 and blows up as x->+infinity, y=-(3^x) must be a reflection of y=(3^x) about the x-axis.

To clarify this, as x-> -infinity, y=-(3^x) approaches zero from the negative half of the y-axis, it crosses y=-1 at x=0 and it goes to some large negative value as x gets bigger and bigger in the positive direction.

In this context, the value a=-1 may be understood as scaling y=(b^x) by 1 (no scaling at all) and taking the opposite polarity (or sign).

Next, consider another possibility, y=a*(-b)^x.
Like before, a is simply a scaling factor, which changes the vertical scale. It doesn't do anything spectacular, except changing the overall shape of the exponential function by a constant amount. For the purpose of discussion, we can neglect the factor "a" and concentrate on the expression, y=(-b)^x.

The most pressing issue is whether y=(-b)^x is defined in general. Can we really call it an exponential function?

(Case 1) If b is a negative quantity, we get back the positive exponential function.
eg., y=(-(-2))^x is same as y=2^x.

(Case 2) If b is a positive quantity, then, we are raising a negative number (inside the parenthesis) to some power of x.

(sub-case a) If x is restricted to the set of natural numbers, N = {1,2,3,4,5,...}, then, this function is well defined. eg., if we have b=2, then, y=(-b)^x yields
y(1)=-2,y(2)=4,y(3)=-8,y(4)=16,y(5)=-32 and so on.
We get an oscillatory waveform, where the sign is negative if integer x is odd, the sign is positive, if integer x is even.

(sub-case b) If x is some positive rational value.
Consider what happens when x=-2 and 1/2 in the previous example. When x=-2, we have y=(-2)^(-2)=1/[(-2)^2]=1/4, this is okay, we are taking the reciprocal of (-2)^2, that's all. But when x=1/2, y=(-2)^(1/2)=sqrt(-2) is the same as taking the square root of a negative number. y is undefined! We get into trouble. In fact, y=(-2)^x is undefined when x=1/(2k), for non-zero integers "k".

Summary:
(1) When we consider a continuous exponential function, where x can take any positive or negative real values, the exponential function is only defined when the base (-b) is positive in y=a*(-b)^x. Or if you prefer, the base c is positive in y=a*(c)^x.
(2) "a" is nothing more than a scaling factor, which stretches or shrinks the vertical scale by some constant amount.
(3) If "a" is negative, the function y=a*(-b)^x is simply a reflection of y=(-a)*(-b)^x about the horizontal axis.

(4) However, if we only allow x to assume integer values, then, a discrete exponential function results. It is only defined at points where x is an integer. We cannot say anything about the function y[x] when x is a non-integer.

[a] In this case, we get an oscillatory exponential function y=(-b)^x, which has the same magnitude envelop as the y=(b)^x, except that the sign is negative whenever the integer x is odd.
[b] Furthermore, the y-value when x is negative is the reciprocal of the corresponding y-value when x is positive.
eg., for y=(-3)^(3)=(-1*3)^(3)=[(-1)^3]*[3^3]=-1*[3^3]=-27.
        y=(-3)^(-3)=...skip a few steps...=-1*[3^(-3)]=-1/[3^3]=-1/27

What is y=(-3)^(1/3) by the way?
y=cube_root(-3)= -1 * cube_root_of(3).
Can easily check by taking the inverse of cube root, i.e., cubing everything to see that (-1)^3 * [cube_root(3)]^3 = -3

Get back to me if anything is unclear.

Cheers.