Chapter 3.1, Problem 12E

Solution

To find: a formula for the exponential function whose values are given in the table.

  $g\left(x\right)=12\cdot {\left(\frac{1}{3}\right)}^x$

Given information:

The table giving the values of the function is:

x	f ( x )
-2	108
-1	36
0	12
1	4
2	$\frac{4}{3}$

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 table given above gives various values of the given function. One such value is: When $x=0$, $g\left(x\right)=12$

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

  $\begin{array}{c}g\left(0\right)=a\cdot b^0\\ 12=a\cdot 1\left[b^0=1\right]\\ 12=a\end{array}$

This gives that,

  $g\left(x\right)=12\cdot b^x$   ..... (2)

Now, choose another value from the given table.

When $x=1$, $g\left(x\right)=4$

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

  $\begin{array}{c}g\left(1\right)=12\cdot b^1\\ 4=12b\\ b=\frac{4}{12}\\ b=\frac{1}{3}\end{array}$

This gives that,

  $g\left(x\right)=12\cdot {\left(\frac{1}{3}\right)}^x$