To determine which of the values for [tex]\( P \)[/tex] and [tex]\( a \)[/tex] will cause the function [tex]\( f(x) = P \cdot a^x \)[/tex] to be an exponential growth function, we need to understand the conditions for exponential growth.
For a function [tex]\( f(x) = P \cdot a^x \)[/tex] to represent exponential growth:
- The base [tex]\( a \)[/tex] must be greater than 1.
- The parameter [tex]\( P \)[/tex] can be any non-zero constant.
Let's examine each option by checking the value of [tex]\( a \)[/tex]:
Option A: [tex]\( P = 5; a = 1 \)[/tex]
- Here, [tex]\( a = 1 \)[/tex].
- Since [tex]\( a \)[/tex] is not greater than 1, this function will not exhibit exponential growth.
Option B: [tex]\( P = \frac{1}{5}; a = 2 \)[/tex]
- Here, [tex]\( a = 2 \)[/tex].
- Since [tex]\( a > 1 \)[/tex], this function will exhibit exponential growth regardless of the value of [tex]\( P \)[/tex], as long as [tex]\( P \neq 0 \)[/tex].
Option C: [tex]\( P = \frac{1}{5}; a = \frac{1}{2} \)[/tex]
- Here, [tex]\( a = \frac{1}{2} \)[/tex].
- Since [tex]\( a < 1 \)[/tex], this function will represent exponential decay, not growth.
Option D: [tex]\( P = 5; a = \frac{1}{2} \)[/tex]
- Here, [tex]\( a = \frac{1}{2} \)[/tex].
- Since [tex]\( a < 1 \)[/tex], this function will also represent exponential decay, not growth.
After evaluating all options, we conclude that the function [tex]\( f(x) = P \cdot a^x \)[/tex] will be an exponential growth function for:
Option B: [tex]\( P = \frac{1}{5}; a = 2 \)[/tex]
