To determine the correct relationship between the number of visitors each year, let's carefully analyze the function provided:
[tex]\[ f(x) = 12,419 \cdot (1.4)^x \][/tex]
This function represents the number of visitors to a website [tex]\( x \)[/tex] years after it was launched.
The general form of an exponential function is:
[tex]\[ f(x) = A \cdot b^x \][/tex]
where:
- [tex]\( A \)[/tex] is the initial amount or starting value when [tex]\( x = 0 \)[/tex].
- [tex]\( b \)[/tex] is the base and represents the growth factor per unit time (in this case, per year).
Given [tex]\( f(x) = 12,419 \cdot (1.4)^x \)[/tex]:
- [tex]\( A = 12,419 \)[/tex], which indicates the number of visitors at [tex]\( x = 0 \)[/tex], or the initial number of visitors.
- [tex]\( b = 1.4 \)[/tex], which is the growth factor per year.
The growth factor, [tex]\( b \)[/tex], indicates how the number of visitors changes each year. Specifically:
- If [tex]\( b > 1 \)[/tex], it means the quantity is increasing each year.
- The value of [tex]\( b \)[/tex] tells us the factor by which the number of visitors is multiplied each year.
In our case, [tex]\( b = 1.4 \)[/tex], which means each year, the number of visitors is multiplied by 1.4 from the previous year.
Breaking down the options:
- 4 times: This would mean the number of visitors is multiplied by 4 each year, which is not correct.
- 4 more than: This would imply the number of visitors increases by an additional 4 visitors each year, which is not consistent with multiplication.
- 0.4 times: This would suggest a decrease (multiplying by less than 1), which is also incorrect.
- 1.4 times: This correctly indicates that each year's visitors are 1.4 times the number from the previous year.
Thus, the number of visitors each year is multiplied by:
[tex]\[ \boxed{1.4 \text{ times}} \][/tex]
This corresponds to option D.
