Question

Find an exponential function to model the data. xy172163304615124627175221) f(x) = 116.4 – 42.8 ln x2) f(x) = 2.04(3.56)x3) f(x) = 3.56(2.04)x4) f(x) = –42.8 + 116.4 ln x

Answer 1

From the given table, let's find an exponential function to model the data.

To write the exponential function, apply the formula:

`f(x)=ab^x`

Where:

b is the rate of change.

We have:

`f(x)=7=ab^1`

Now substitute (7, 522) for the values of x and f(x):

`522=ab^7`

Divide both equations to find b:

`\begin{gathered} (ab^7)/(ab^1)=(522)/(7) \\ \\ b^6=74.57 \\ \\ b=2.04 \end{gathered}`

The value of b is 2.04.

To find the value of a, we have:

`f(x)=a(2.04)^x`

Substituet (7, 522) for values of x and f(x):

`\begin{gathered} 522=a(2.04)^7 \\ \\ 522=147.032a \end{gathered}`

Divide both sides by 147.032a:

`\begin{gathered} (522)/(147.032)=(147.032a)/(147.032) \\ \\ 3.56=a \\ \\ a=3.56 \end{gathered}`

Therefore, the exponential function to model the data is:

`f(x)=3.56(2.04)^x`

ANSWER:

`f(x)=3.56(2.04)^x`