Chapter 3.2, Problem 20E

Solution

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

  $g\left(x\right)=-5.8\cdot {0.8}^x$

Given information:

The table containing the values of the given function:

x	g ( x )
-2	-9.0625
-1	-7.25
0	-5.8
1	-4.64
2	-3.7123

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 $x=0$, $g\left(x\right)=-5.8$

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

  $\begin{array}{c}g\left(0\right)=a\cdot b^0\\ -5.8=a\cdot 1\left[b^0=1\right]\\ -5.8=a\end{array}$

This gives that,

  $g\left(x\right)=-5.8\cdot b^x$   ..... (2)

Now, choose another value from the given table.

When $x=1$, $g\left(x\right)=-4.64$

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

  $\begin{array}{c}g\left(1\right)=-5.8\cdot b^1\\ -4.64=-5.8\cdot b\\ b=\frac{-4.64}{-5.8}\\ b=0.8\end{array}$

This gives that,

  $g\left(x\right)=-5.8\cdot {0.8}^x$