Respuesta :
Given:
f(x) is an exponential function.
f(4) = 7.
f(5.5) = 53.
Since f(x) is an exponential function, f(x) can be expressed as,
[tex]f(x)=ab^x[/tex]Here, a and b are constants.
Hence, we can write
[tex]\begin{gathered} f(4)=ab^4----(1) \\ f(5.5)=ab^{5.5}\text{ ------(2)} \end{gathered}[/tex]Divide equation (2) by (1).
[tex]\frac{f(5.5)}{f(4)}=\frac{ab^{5.5}}{ab^4}[/tex]Substitute f(4) = 7 and f(5.5) = 53 in the above equation.
[tex]\begin{gathered} \frac{53}{7}=\frac{b^{5.5}^{}}{b^4} \\ \frac{53}{7}=b^{5.5-4} \\ \frac{53}{7}=b^{1.5} \\ \frac{53}{7}=b^{\frac{3}{2}} \\ b=(\frac{53}{7})^{\frac{2}{3}} \\ b=3.855 \end{gathered}[/tex]Now, the value of a can be obtained as,
[tex]\begin{gathered} f(4)=ab^4 \\ 7=a(\frac{53}{7})^{\frac{2}{3}} \\ a=\frac{7}{(\frac{53}{7})^{\frac{2}{3}}} \\ =1.815 \end{gathered}[/tex]