Question

If f(x) is an exponential functionwhere f(-1) = 26 andf(6.5) = 14, then find the value off(5), to the nearest hundredth.

Answer 1

The general form of an exponential function is:

`y=a\cdot b^x^{}`

where a ≠ 0, b > 0 and b ≠ 1

f(-1) = 26 and f(6.5) = 14, that is,

`\begin{gathered} 26=a\cdot b^(-1) \\ 26=(a)/(b) \\ 26\cdot b=a \end{gathered}`
`\begin{gathered} 14=a\cdot b^(6.5) \\ 14=26\cdot b\cdot b^(6.5) \\ 14=26\cdot b^(7.5) \\ (14)/(26)=b^(7.5) \\ \ln ((14)/(26))=7.5\cdot\ln b \\ -(0.619)/(7.5)=\ln b \\ e^(-0.0825)=b \\ 0.92=b \end{gathered}`

Then,

`\begin{gathered} a=26\cdot b \\ a=26\cdot0.92 \\ a=23.92 \end{gathered}`

Finally, the function is:

`y=23.92\cdot0.92^x^{}`