To determine the relationship between the number of weeks [tex]\(x\)[/tex] and the number of views [tex]\(f(x)\)[/tex], we can use an exponential function of the form:
[tex]\[ f(x) = a \cdot b^x \][/tex]
Here, [tex]\(a\)[/tex] represents the initial number of views when [tex]\(x = 0\)[/tex], and [tex]\(b\)[/tex] represents the growth factor per week.
Based on the given data and calculations, we find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
From the table:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Weeks}, \; x & \text{Views}, \; f(x) \\
\hline 0 & 5{,}120 \\
\hline 1 & 6{,}400 \\
\hline 2 & 8{,}000 \\
\hline 3 & 10{,}000 \\
\hline 4 & 12{,}500 \\
\hline 5 & 15{,}625 \\
\hline
\end{array}
\][/tex]
We can determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:
- [tex]\(a\)[/tex] is the initial value when [tex]\(x = 0\)[/tex], which is [tex]\(5{,}120\)[/tex].
- [tex]\(b\)[/tex] is the growth rate, which is approximately [tex]\(1.25\)[/tex].
Thus, substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the exponential function:
[tex]\[ f(x) = 5119.999999999992 \cdot (1.2499999999999998)^x \][/tex]
For simplicity, we can round the coefficients to:
[tex]\[ f(x) = 5120 \cdot 1.25^x \][/tex]
This equation models the relationship between the number of weeks [tex]\(x\)[/tex] and the number of views [tex]\(f(x)\)[/tex]:
[tex]\[ \boxed{f(x) = 5120 \cdot 1.25^x} \][/tex]
