3e^2x = G

-3e^2x = G

-3e^-2x = D

1/8e^2x = G

-1/8e^-2x = D

Would that be correct? I just made these examples up in my head.

How can you tell if its growth or decay? I just look at the exponents. If is a negative then its decay? if its positive then its growth?

The coefficient (which represents the original amount) can't be negative for growth or decay. Otherwise, you're correct.

Ahh thanks. I see. Hey would you happen to know how to graph log functions? and find the domain and range for them?

Since logs (which are just exponents) will never cause a number to be 0 or negative (you can't raise e, for example, to any exponent that will yield a value of 0 or below) the values of x that DO cause the number to be 0 or less will create an vertical asymptote.

EXAMPLE:

f(x)=log(6x+2)-3

You know that the part in the parenthesis must be greater than 0, so find the value of x that cause it to be 0, and say that the domain is all numbers greater than that value for x.
( DOMAIN: x > -1/3 )

The range is all real numbers, because y could be any positive or negative number (the log part will yield a negative number when there's a number that's less than 1 in the parenthesis and a positive value when the part in the parenthesis is greater than 1).

By the way, at the vertical asymptote where x = -1/3, the value for y gets infinitely negative as you get closer and closer to that line (since the values in the parenthesis are getting smaller and smaller as you approach the minimum value for x, -1/3). Remember that it's FRACTIONS that cause logs to be negative (like getting from 2 to 1/2 requires an exponent [or log] of -1 and 2 to 1/4 requires an exponent [log] of -2). LOGS ARE EXPONENTS! Are you getting it now?

I remember my teacher saying that the Domain and Range are opposite, which did that apply to?

There's a constraint in the Domain with logarithms where the variable is in the parenthesis (which must be greater than 0). The Range is all reals with logarithms when the variable is in the parenthesis.

There's a constraint in the Range when the variable is in the exponent (the output will always be greater than 0). The Domain is all reals when the variable is the exponent.

To determine whether a given equation represents growth or decay, you can indeed look at the exponent. For exponential functions of the form f(x) = a * e^(bx), where a and b are constants, the sign of the exponent, b, can indicate whether it represents growth or decay.

If the exponent, b, is positive (b > 0), then the function represents growth. As the value of x increases, the function value will also increase.

If the exponent, b, is negative (b < 0), then the function represents decay. As the value of x increases, the function value will decrease.

Now, let's go through the examples you provided and determine whether they represent growth or decay.

1. 3e^(2x) = G: The exponent, 2x, is positive, so this equation represents growth.

2. -3e^(2x) = G: The exponent, 2x, is still positive, so this equation also represents growth. The negative sign does not affect whether it is growth or decay; it only indicates a negative output value.

3. -3e^(-2x) = D: The exponent, -2x, is negative, so this equation represents decay.

4. 1/8e^(2x) = G: Again, the exponent, 2x, is positive, so this equation represents growth. The coefficient 1/8 only scales the magnitude but does not change the nature of growth.

5. -1/8e^(-2x) = D: Similarly, the exponent, -2x, is negative, meaning this equation represents decay. The coefficient -1/8 scales the magnitude, but it still represents decay.

Therefore, in your examples, equations 1, 2, 4 represent growth, while equations 3 and 5 represent decay.