Chapter 3.1, Problem 13E

Solution

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

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

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 $x=0$, $f\left(x\right)=3$

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

  $\begin{array}{c}f\left(0\right)=a\cdot b^0\\ 3=a\cdot 1\left[b^0=1\right]\\ 3=a\end{array}$

This gives that,

  $f\left(x\right)=3\cdot b^x$   ..... (2)

Now, choose another value from the given table.

When $x=2$, $f\left(x\right)=6$

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

  $\begin{array}{c}f\left(2\right)=3\cdot b^2\\ 6=3b^2\\ b^2=2\\ b=2^{\frac{1}{2}}\end{array}$

Since, b is a positive real number, so here the negative value of square root is discarded.

This gives that,

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