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Which of these values for [tex]\(P\)[/tex] and [tex]\(a\)[/tex] will cause the function [tex]\(f(x) = Pa^x\)[/tex] to be an exponential growth function?

A. [tex]\(P = 6\)[/tex]; [tex]\(a = \frac{1}{8}\)[/tex]

B. [tex]\(P = 6\)[/tex]; [tex]\(a = 1\)[/tex]

C. [tex]\(P = \frac{1}{6}\)[/tex]; [tex]\(a = 8\)[/tex]

D. [tex]\(P = \frac{1}{6}\)[/tex]; [tex]\(a = \frac{1}{8}\)[/tex]



Answer :

GinnyAnswer
To determine which option will result in the function [tex]\( f(x) = Pa \)[/tex] being an exponential growth function, we need to understand the characteristics of exponential growth. Exponential growth occurs when the base of the exponential function, given as [tex]\( a \)[/tex], is greater than 1.

Let's examine each option:

1. Option A: [tex]\( P = 6 \)[/tex], [tex]\( a = \frac{1}{8} \)[/tex]
- Here, [tex]\( a = \frac{1}{8} \)[/tex]. Since this value is less than 1, it represents exponential decay, not exponential growth.

2. Option B: [tex]\( P = 6 \)[/tex], [tex]\( a = 1 \)[/tex]
- Here, [tex]\( a = 1 \)[/tex]. When [tex]\( a = 1 \)[/tex], the function does not represent exponential growth or decay because it remains constant. Thus, [tex]\( a = 1 \)[/tex] does not qualify for exponential growth.

3. Option C: [tex]\( P = \frac{1}{6} \)[/tex], [tex]\( a = 8 \)[/tex]
- Here, [tex]\( a = 8 \)[/tex]. Since this value is greater than 1, it represents exponential growth. Therefore, this option is valid for exponential growth.

4. Option D: [tex]\( P = \frac{1}{6} \)[/tex], [tex]\( a = \frac{1}{8} \)[/tex]
- Here, [tex]\( a = \frac{1}{8} \)[/tex]. Since this value is less than 1, it represents exponential decay, not exponential growth.

Thus, the option that results in the function being an exponential growth function is:

C. [tex]\( P = \frac{1}{6} \)[/tex]; [tex]\( a = 8 \)[/tex].

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