Question

If f(x) is an exponential function where f(1.5) = 7 and f(8.5) = 46, then find the value of f(14), to the nearest hundredth.

Answer 1

`f(14)=201.932`

Explanation:

Set up a system of equations and solve for "b"

`\left \{ {{46=ab^(8.5)} \atop {7=ab^(1.5)}} \right\\ \\\left \{ {{(46)/(b^(8.5))=a } \atop {(7)/(b^(1.5))=a} \right\\\\(46)/(b^(8.5))=(7)/(b^(1.5))\\ \\46b^(1.5)=7b^(8.5)\\\\46=7b^7\\\\(46)/(7)=b^7\\ \\b=1.308604899`

Determine the value of "a" using "b"

`y=ab^x\\\\46=a(1.308604899)^(8.5)\\\\46=9.837224985a\\\\a=4.676115477`

Find f(14) using the new function

`f(x)=4.676115477(1.308604899)^x\\\\f(14)=4.676115477(1.308604899)^(14)\\\\f(14)\approx201.932`