Question

Find the equation for the exponential function that passes through the points (4, 3) and (7, 8).

Answer 1

Given:

The exponential function is expressed as,

`f(x)=ab^x`

Put the point (4,3) in the above function,

`\begin{gathered} \text{Let,y}=ab^x \\ 3=ab^4\ldots\ldots\ldots.\ldots..(1) \end{gathered}`

Put the point (7,8),

`8=ab^7\ldots.\ldots(2)`

Solve equation (1) for a,

`\begin{gathered} 3=ab^4 \\ a=3b^(-4) \\ \text{Put it in equation (2)} \\ 8=ab^7 \\ 3b^(-4)b^7=8 \\ b^3=(8)/(3) \\ b=\frac{2}{\sqrt[3]{3}} \end{gathered}`

The equation (1) becomes,

`\begin{gathered} 3=ab^4 \\ 3=a(\frac{2}{\sqrt[3]{3}})^4 \\ 3=a*\frac{2^4}{3^{(4)/(3)}} \\ a=\frac{3*3^{(4)/(3)}}{2^4} \\ a=0.8113 \end{gathered}`

So, the exponential function is,

`\begin{gathered} f(x)=ab^x \\ f(x)=0.8113(\frac{2}{\sqrt[3]{3}})^x \\ f(x)=0.8113(1.3867)^x \end{gathered}`

Answer:

`f(x)=0.8113(1.3867)^x`