To determine which value 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 criteria for exponential growth. For the function to represent exponential growth, the base of the exponent, [tex]\( a \)[/tex], must be greater than 1. This is because when [tex]\( a > 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( a^x \)[/tex] grows exponentially.
Let's analyze each option:
### Option A: [tex]\( P = \frac{1}{13} \)[/tex] and [tex]\( a = 6 \)[/tex]
- Here, [tex]\( a = 6 \)[/tex], which is greater than 1.
- Since [tex]\( a > 1 \)[/tex], this will cause [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function.
### Option B: [tex]\( P = 13 \)[/tex] and [tex]\( a = \frac{1}{6} \)[/tex]
- Here, [tex]\( a = \frac{1}{6} \)[/tex], which is less than 1.
- Since [tex]\( a < 1 \)[/tex], this will not cause [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function.
### Option C: [tex]\( P = \frac{1}{13} \)[/tex] and [tex]\( a = \frac{1}{6} \)[/tex]
- Here, [tex]\( a = \frac{1}{6} \)[/tex], which is less than 1.
- Since [tex]\( a < 1 \)[/tex], this will not cause [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function.
### Option D: [tex]\( P = 13 \)[/tex] and [tex]\( a = 1 \)[/tex]
- Here, [tex]\( a = 1 \)[/tex].
- Since [tex]\( a = 1 \)[/tex], [tex]\( f(x) = P a^x \)[/tex] becomes a constant function instead of an exponential growth function.
Based on this analysis, the only option that meets the criteria for exponential growth where [tex]\( a > 1 \)[/tex] is:
Option A: [tex]\( P = \frac{1}{13} \)[/tex] and [tex]\( a = 6 \)[/tex]
This is the option that will cause the function [tex]\( f(x) = P a^x \)[/tex] to be an exponential growth function.
