Chapter 3, Problem 4RE

Solution

To find: a formula for the exponential function whose graph is given.

  $f\left(x\right)=2\cdot {\left(\frac{1}{2}\right)}^{\frac{x}{3}}$

Given information:

The graph of the required function 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:

The graph given above gives various values of the given function. One such value is: When $x=0$, $f\left(x\right)=2$

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

  $\begin{array}{c}f\left(0\right)=a\cdot b^0\\ 2=a\cdot 1\left[b^0=1\right]\\ 2=a\end{array}$

This gives that,

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

Now, choose another value from the given table.

When $x=3$, $f\left(x\right)=1$

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

  $\begin{array}{c}f\left(3\right)=2\cdot b^3\\ 1=2b^3\\ b^3=\frac{1}{2}\\ b={\left(\frac{1}{2}\right)}^{\frac{1}{3}}\end{array}$

This gives that,

  $\begin{array}{l}f\left(x\right)=2\cdot {\left({\left(\frac{1}{2}\right)}^{\frac{1}{3}}\right)}^x\\ f\left(x\right)=2\cdot {\left(\frac{1}{2}\right)}^{\frac{x}{3}}\end{array}$