Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear

function that models the data; for those that are exponential, find an exponential function that models the data.

x	f(x)
-1	$\frac{3}{2}$
0	3
1	6
2	12
3	24

Step-by-Step Solution

Answer

This is not a linear but exponential function where

$f\left(x\right)=3.2^x$

Step 1. Given

A table with values of $x,f\left(x\right)$ to trace the function.

Step 2. Write the concept

We will check for domain and range that will gives us a clue about exponentiality or linearity of the function.

Step 3. Calculation

Here the change in domain is just 1 unit but change in range is of higher order.

Also the change is not constant therefore, it is not linear.

For exponential function we choose a value from domain i.e. $1\&-1$ but the values at this domain is reciprocal of each other therefore, it is exponential.

Consider

$$\begin{gathered} \frac{f\left(x\right)}{3}=a^x \\ Now at x=1 we have \\ \frac{f\left(x\right)}{3}=2 \\ Now 2=a^1 \\ Thus the function is \\ \frac{f\left(x\right)}{3}=2^x \\ f\left(x\right)=2^x.3 \end{gathered}$$