Option C: [tex]$f(x)=3^{5}\left(\frac{1}{3}\right)^{x}$[/tex] is the function that represent exponential decay.
Explanation:
The exponential decay can be represented by
[tex]$f(x)=a \cdot b^{x}$[/tex] where [tex]a>0[/tex] and [tex]0<b<1[/tex]
Option A: [tex]$f(x)=3(1.7)^{x-2}$[/tex]
From the function, we can see that [tex]a=3[/tex] and [tex]b=1.7[/tex]
Thus, [tex]a>0[/tex] and [tex]b>1[/tex]
Thus, the function [tex]$f(x)=3(1.7)^{x-2}$[/tex] does not represent exponential decay.
Hence, Option A is not the correct answer.
Option B: [tex]$f(x)=3(1.7)^{-2 x}$[/tex]
From the function, we can see that [tex]a=3[/tex] and [tex]b=1.7[/tex]
Thus, [tex]a>0[/tex] and [tex]b>1[/tex]
Thus, the function [tex]$f(x)=3(1.7)^{-2 x}$[/tex] does not represent exponential decay.
Hence, Option B is not the correct answer.
Option C: [tex]$f(x)=3^{5}\left(\frac{1}{3}\right)^{x}$[/tex]
From the function, we can see that [tex]a=3^5=243[/tex] and [tex]b=\frac{1}{3} =0.3333[/tex]
Thus, [tex]a>0[/tex] and [tex]0<b<1[/tex]
Thus, the function [tex]$f(x)=3^{5}\left(\frac{1}{3}\right)^{x}$[/tex] represent exponential decay.
Hence, Option C is the correct answer.
Option D: [tex]$f(x)=3^{5}(2)^{-x}$[/tex]
From the function, we can see that [tex]a=3^5=243[/tex] and [tex]b=2[/tex]
Thus, [tex]a>0[/tex] and [tex]b>1[/tex]
Thus, the function [tex]$f(x)=3^{5}(2)^{-x}$[/tex] does not represent exponential decay.
Hence, Option D is not the correct answer.
