Chapter 3.2, Problem 21E

Solution

To find: a formula for the exponential function whose graph is shown in the figure.

  $f\left(x\right)\approx 4\cdot {1.15}^x$

Given information:

The given graph is:

  

Definition Used:

The general form of an exponential function f ( x ) is

  $f\left(x\right)=a\cdot b^x$   ..... (1)

Where $a\ne 0$, b is a positive real number and $b\ne 1$ .

Explanation:

Substitute the values from the graph in the general form to find the values of the constants.

One such value is: When $x=0$, $f\left(0\right)=4$

Substituting this in the general form (1) of the exponential function.

  $\begin{array}{c}f\left(0\right)=a\cdot b^0\\ 4=a\cdot 1\left[b^0=1\right]\\ 4=a\end{array}$

This gives that,

  $f\left(x\right)=4\cdot b^x$   ..... (2)

Now, choose another value from the given table.

When $x=5$, $f\left(x\right)=8.05$

Substituting this in the general form (1) of the exponential function.

  $\begin{array}{c}f\left(5\right)=2.3\cdot b^5\\ 8.05=4\cdot b^5\\ b^5=\frac{8.05}{4}\end{array}$

  $\begin{array}{c}b=\sqrt[5]{\frac{8.05}{4}}\\ b\approx 1.15\end{array}$

This gives that,

  $f\left(x\right)\approx 4\cdot {1.15}^x$