Given the exponential function A(x) = P(1 + r)x, what value for r will make the function a decay function?
r = −2.1
r = 2.1
r = 0
r = 0.1
To determine whether the exponential function A(x) = P(1 + r)x is a decay function or a growth function, we need to examine the value of the growth factor, which is (1 + r).
In a decay function, the growth factor must be less than 1. To represent decay, the value of r should be negative.
Let's check the given values of r:
1. r = -2.1: Since r is negative, the growth factor is (1 - 2.1) = -1.1, which is less than 1. Therefore, A(x) = P(1 - 2.1)x represents a decay function.
2. r = 2.1: Since r is positive, the growth factor is (1 + 2.1) = 3.1, which is greater than 1. Therefore, A(x) = P(1 + 2.1)x does not represent a decay function.
3. r = 0: The growth factor is (1 + 0) = 1, which is not less than 1. Therefore, A(x) = P(1 + 0)x does not represent a decay function.
4. r = 0.1: Since r is positive, the growth factor is (1 + 0.1) = 1.1, which is greater than 1. Therefore, A(x) = P(1 + 0.1)x does not represent a decay function.
Based on the analysis, the only value of r that makes the function a decay function is r = -2.1.
Think of the graphs of e^x and e^-x
e^-x is a decay function, since
e^-x = (1/e)^x
A decay function a^x must have 0<a<1.
so, you need
0 < 1+r < 1
-1 < r < 0
None of the choices qualifies. a^x is not a real function if a<0.
I suspect a typo.