Q. 32 - 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	$\frac{1}{2}$
0	$\frac{1}{4}$
1	$\frac{1}{8}$
2	$\frac{1}{16}$
3	$\frac{1}{32}$

Step-by-Step Solution

Answer

This is not a linear but exponential function where

$g\left(x\right)=\frac{1}{4}{\left(\frac{1}{2}\right)}^x$

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

$$\begin{gathered} g\left(x\right)=c{a}^x \\ =\frac{1}{4}{\left(\frac{1}{2}\right)}^x \end{gathered}$$