Chapter 3.1, Problem 11E

Solution

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

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

Given information:

The table giving the values of the function is:

x	f ( x )
-2	6
-1	3
0	$\frac{3}{2}$
1	$\frac{3}{4}$
2	$\frac{3}{8}$

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 $x=0$, $f\left(x\right)=\frac{3}{2}$

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

  $\begin{array}{c}f\left(0\right)=a\cdot b^0\\ \frac{3}{2}=a\cdot 1\left[b^0=1\right]\\ \frac{3}{2}=a\end{array}$

This gives that,

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

Now, choose another value from the given table.

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

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

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

This gives that,

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