Chapter 3.2, Problem 19E

Solution

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

  $f\left(x\right)=2.3\cdot {1.25}^x$

Given information:

The table containing the values of the given function:

x	f ( x )
-2	1.472
-1	1.84
0	2.3
1	2.875
2	3.59375

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 $x=0$, $f\left(x\right)=2.3$

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

  $\begin{array}{c}f\left(0\right)=a\cdot b^0\\ 2.3=a\cdot 1\left[b^0=1\right]\\ 2.3=a\end{array}$

This gives that,

  $f\left(x\right)=2.3\cdot b^x$   ..... (2)

Now, choose another value from the given table.

When $x=1$, $f\left(x\right)=2.875$

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

  $\begin{array}{c}f\left(1\right)=2.3\cdot b^1\\ 2.875=2.3\cdot b\\ b=\frac{2.875}{2.3}\\ b=1.25\end{array}$

This gives that,

  $f\left(x\right)=2.3\cdot {1.25}^x$