Chapter 7.7, Problem 42PPE

To determine

To calculate: two exponential growth functions $f\left(x\right)$ and $g\left(x\right)$ such that $f\left(x\right)<g\left(x\right)$ for $x<3$ and $f\left(x\right)>g\left(x\right)$ for $x>3$

Answer to Problem 42PPE

  $f\left(x\right)=5\left(3^x\right)$ and $g\left(x\right)=5\left(5^x\right)$

Explanation of Solution

Given information:

Two exponential growth functions $f\left(x\right)$ and $g\left(x\right)$ such that $f\left(x\right)<g\left(x\right)$ for $x<3$ and $f\left(x\right)>g\left(x\right)$ for $x>3$

Calculation:

Let, $f\left(x\right)=5\left(3^x\right)$ and $g\left(x\right)=5\left(5^x\right)$

For $x<3$

For $x=2$

  $\begin{array}{l}f\left(2\right)=5\left(3^2\right)=5\left(9\right)=45\\ g\left(2\right)=5\left(5^2\right)=5\left(25\right)=125\\ f\left(x\right)<g\left(x\right)\end{array}$

For $x=1$

  $\begin{array}{l}f\left(1\right)=5\left(3^1\right)=5\left(3\right)=15\\ g\left(1\right)=5\left(5^1\right)=5\left(5\right)=25\\ f\left(x\right)<g\left(x\right)\end{array}$

For $x>3$

For $x=4$

  $\begin{array}{l}f\left(4\right)=5\left(3^4\right)=5\left(81\right)=405\\ g\left(4\right)=5\left(5^4\right)=5\left(625\right)=3125\\ g\left(x\right)>f\left(x\right)\end{array}$

For $x=5$

  $\begin{array}{l}f\left(5\right)=5\left(3^5\right)=5\left(243\right)=1215\\ g\left(5\right)=5\left(5^5\right)=5\left(3125\right)=15625\\ g\left(x\right)>f\left(x\right)\end{array}$