Chapter 6.3, Problem 28E

To determine

To graph: Linear or exponential function.

Answer to Problem 28E

Linear or exponential function is graphed with the given conditions.

Explanation of Solution

Given information :

  $f(x)>g(x)$ where $0<x<4$

  $f(x)<g(x)$ where $x<0\&x>4$

Concept used:

Use patterns between consecutive data pairs to determine type of function.

Linear function: - The difference of consecutive y-values is constant.

Exponential function: - Consecutive y-values have a common ratio. In each case, the differences of consecutive x-values need to be constant.

We have to determine a linear function $f$ and an exponential function $g$ :

  $f(x)>g(x)$ where $0<x<4$

  $f(x)<g(x)$ where $x<0\&x>4$

The two functions should intersect in $x=0\&x=4$ .

Let’s consider the functions:

  $\begin{array}{l}f(x)=mx+n\\ g(x)=a{b}^x\end{array}$

We have:

  $\begin{array}{l}m(0)+n=a{b}^0\\ m(4)+n=a{b}^4\\ n=a\\ 4m+n=a{b}^4\end{array}$

  $\begin{array}{l}a=a{b}^4-4m\\ m=\frac{a{b}^4-a}{4}\end{array}$

Therefore, the functions are:

  $\begin{array}{l}f(x)=\frac{a{b}^4-a}{4}x+a\\ g(x)=a{b}^x\end{array}$

For example, for $a=1,b=3$ we have:

  

Hence,

Linear or exponential function is graphed with the given conditions.