The given function is:
[tex]f(x)=Ca^x[/tex]
It is given that its graph passes through the points (2,9.375) and (5,18.311).
Recall that if the graph of a function passes through a point, then the coordinates of the point satisfy the equation of the function.
Substitute x=2 and f(x)=9.375 into the equation of the function:
[tex]9.375=Ca^2[/tex]
Substitute x=5 and f(x)=18.311 into the equation of the function:
[tex]18.311=Ca^5[/tex]
Hence, the system of equation is:
[tex]\begin{gathered} 9.375=Ca^2 \\ 18.311=Ca^5 \end{gathered}[/tex]
Solve the system of equations to find the constants C and a.
Divide the second equation by the first equation:
[tex]\begin{gathered} \frac{18.311}{9.375}=\frac{Ca^5}{Ca^2} \\ \Rightarrow\frac{18.311}{9.375}=\frac{\cancel{C}a^5}{\cancel{C}a^2}\Rightarrow\frac{18.311}{9.375}=\frac{a^5}{a^2} \end{gathered}[/tex]
Solve for a in the equation:
[tex]\begin{gathered} \text{Swap the sides of the equation:} \\ \frac{a^5}{a^2}=\frac{18.311}{9.375}\Rightarrow a^{5-2}=1.9532 \\ \Rightarrow a^3=1.9532 \\ \Rightarrow a=\sqrt[3]{1.9532}\approx1.25 \end{gathered}[/tex]
Substitute this value of a into the first equation to solve for C:
[tex]\begin{gathered} 9.375=C(1.25)^2 \\ \Rightarrow C=\frac{9.375}{1.25^2}=6 \end{gathered}[/tex]
Substitute the values of a and C into the initial equation of the function:
[tex]\begin{gathered} f(x)=Ca^x;a=1.25,c=6 \\ \Rightarrow f(x)=6(1.25)^x \end{gathered}[/tex]
The required formula for f(x) is:
[tex]f(x)=6(1.25)^x[/tex]
