To determine which of these values for [tex]\( P \)[/tex] and [tex]\( a \)[/tex] will cause the function [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function, we need to recall the definition of an exponential growth function. Specifically, for [tex]\( f(x) = P a^x \)[/tex] to represent exponential growth, the base [tex]\( a \)[/tex] must be greater than 1.
Let's analyze each of the given options:
1. Option A: [tex]\( P = \frac{1}{2} \)[/tex], [tex]\( a = 1 \)[/tex]
- Here, the base [tex]\( a \)[/tex] is 1.
- Since an exponential growth function requires [tex]\( a > 1 \)[/tex], this option does not cause exponential growth because [tex]\( a = 1 \)[/tex].
2. Option B: [tex]\( P = 2 \)[/tex], [tex]\( a = 3 \)[/tex]
- Here, the base [tex]\( a \)[/tex] is 3.
- Since 3 is greater than 1 (i.e., [tex]\( a > 1 \)[/tex]), this option does cause exponential growth.
3. Option C: [tex]\( P = \frac{1}{2} \)[/tex], [tex]\( a = \frac{1}{3} \)[/tex]
- Here, the base [tex]\( a \)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
- Since [tex]\(\frac{1}{3}\)[/tex] is less than 1 (i.e., [tex]\( a < 1 \)[/tex]), this option does not cause exponential growth. In fact, it would represent exponential decay.
4. Option D: [tex]\( P = 2 \)[/tex], [tex]\( a = 1 \)[/tex]
- Again, in this case, the base [tex]\( a \)[/tex] is 1.
- This option does not cause exponential growth because [tex]\( a = 1 \)[/tex].
After reviewing each option, we conclude:
- Only option B ([tex]\( P = 2 \)[/tex], [tex]\( a = 3 \)[/tex]) will cause the function [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function.
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
This means the set of values for [tex]\( P \)[/tex] and [tex]\( a \)[/tex] that will cause the function to be an exponential growth function is found in option B.
