To determine which pair of \( P \) and \( a \) will cause the function \( f(x) = P \cdot a^x \) to be an exponential growth function, we need to evaluate the properties of exponential functions.
For a function to be classified as an exponential growth function, the base \( a \) must be greater than 1. An exponential growth function increases as \( x \) increases, and this happens only if \( a > 1 \).
Now, let's evaluate each option:
Option A: \( P = 3 \) and \( a = \frac{1}{4} \)
- Here, the base \( a = \frac{1}{4} \).
- Since \( \frac{1}{4} < 1 \), this function \( f(x) = 3 \cdot (\frac{1}{4})^x \) is not an exponential growth function.
Option B: \( P = 3 \) and \( a = 1 \)
- Here, the base \( a = 1 \).
- Since \( 1 \) is not greater than \( 1 \) (it is equal to 1), the function \( f(x) = 3 \cdot 1^x \) is not an exponential growth function but rather a constant function.
Option C: \( P = \frac{1}{3} \) and \( a = 4 \)
- Here, the base \( a = 4 \).
- Since \( 4 > 1 \), this function \( f(x) = \frac{1}{3} \cdot 4^x \) is an exponential growth function.
Option D: \( P = \frac{1}{3} \) and \( a = \frac{1}{4} \)
- Here, the base \( a = \frac{1}{4} \).
- Since \( \frac{1}{4} < 1 \), this function \( f(x) = \frac{1}{3} \cdot (\frac{1}{4})^x \) is not an exponential growth function.
Therefore, the pair \( P = \frac{1}{3} \) and \( a = 4 \) in Option C is the only one that causes \( f(x) \) to be an exponential growth function.
So, the correct answer is:
C. [tex]\( P = \frac{1}{3} \)[/tex] and [tex]\( a = 4 \)[/tex]
