Question

If f(x) is an exponential function where f(3.5)=12, and f(9)=75 then find the value of f(7.5) to the nearest hundredth?

Answer 1

Given the general exponential function:

`f(x)=A\cdot b^x`

Where A and b are parameters. Then, using the information provided in the problem:

`\begin{gathered} f(3.5)=A\cdot b^(3.5)=12 \\ f(9)=A\cdot b^9=75 \end{gathered}`

Taking the division:

`\begin{gathered} (A\cdot b^9)/(A\cdot b^(3.5))=(75)/(12) \\ b^(5.5)=6.25 \\ \Rightarrow b=1.39542 \end{gathered}`

Now, we use f(9) = 75 to find the parameter A:

`\begin{gathered} f(9)=A\cdot1.39542^9=75 \\ \Rightarrow A=3.7386 \end{gathered}`

The function is:

`f(x)=3.7386\cdot1.39542^x`

We evaluate at x = 7.5:

`\begin{gathered} f(7.5)=3.7386\cdot1.39542^(7.5) \\ \Rightarrow f(7.5)=45.50 \end{gathered}`