Answer:
[tex]f(x) = 63000(0.97)^x[/tex]
Step-by-step explanation:
Given
[tex]f(0) = 63000[/tex] ---x = 0, in 2012
[tex]f(5) = 54100[/tex] -- x = 5, in 2017
Required
Select all possible equations
Because there is a reduction in the population, as time increases; the rate must be less than 1.
An exponential function is represented as:
[tex]f(x) = ab^x[/tex]
Where
[tex]b = rate[/tex]
rate > 1 in options (a) and (b) i.e. 1.03
This implies that (a) and (b) cannot be true
For option (c), we have:
[tex]f(x) = 63000(0.97)^x[/tex]
Set x = 0
[tex]f(0) = 63000(0.97)^0 = 63000*1=63000\\[/tex]
Set x = 5
[tex]f(5) = 63000(0.97)^5 = 63000*0.8587=54098.1 \approx 54100[/tex]
This is true because the calculated values of f(0) and f(5) correspond to the given values
For option (d), we have:
[tex]f(x) = 52477(0.97)^x[/tex]
Set x = 0
[tex]f(0) = 52477(0.97)^0 - 52477* 1 = 52477[/tex]
This is false because the calculated value of f(0) does not correspond to the given value
For option (e), we have:
[tex]f(x) = 63000(0.97)^\frac{1}{5x}[/tex]
Set x = 0
[tex]f(0) = 63000(0.97)^\frac{1}{5*0} = 63000(0.97)^\frac{1}{0} =[/tex]undefined
This is false because the f(x) is not undefined at x = 0
For option (f), we have:
[tex]f(x) = 52477(0.97)^{5x[/tex]
Set x = 0
[tex]f(0) = 52477(0.97)^{5*0} = 52477(0.97)^0 =52477*1= 52477[/tex]
This is false because the calculated value of f(0) does not correspond to the given value
From the computations above, only (c) [tex]f(x) = 63000(0.97)^x[/tex] is true
