Chapter 6, Problem 8CA

To determine

To identify: Each function that represents exponential growth or decay or neither or linear function.

Answer to Problem 8CA

Exponential	Linear	Neither
$\begin{array}{}$ f(x)=-2{(8)}^x->Growth$$ f(x)=\frac{1}{2}{(3)}^x->Growth$$ f(x)=4{(1.6)}^{\frac{x}{10}}->Growth$$ f(x)=3{(\frac{1}{6})}^x->Decay$\end{array}$	$\begin{array}{l}f(x)=15-x\\ f(x)=-3(4x-1-x)\\ f(x)=\sqrt[4]{16}+2x\end{array}$	$\begin{array}{l}f(x)=6x^2+9\\ f(x)=x(18-x)\end{array}$

Explanation of Solution

Given information: Group of functions are given. Analyze the given function and arrange it in the table.

Formula used:

  $\begin{array}{l}Exponential-function\\ f(x)=a{b}^x\\ b>1(Growth)\\ 0<b<1(decay)\end{array}$

Calculation:

Analyze each and every function. For exponential function, compare the given function with general form and check the value of b

  $\begin{array}{l}f(x)=a{b}^x\\ $ f(x)=-2{(8)}^x$b=8>1,Growth\\ \\ $ f(x)=\frac{1}{2}{(3)}^x$b=3>1,Growth\\ \\ $ f(x)=4{(1.6)}^{\frac{x}{10}}$b=1.6>1,Growth\\ \\ $ f(x)=3{(\frac{1}{6})}^x$b=\frac{1}{6}\\ 0<\frac{1}{6}<1,Decay\end{array}$

When the degree of the function is 1 then it is linear

When the degree of the function is 2 then it is quadratic

  $\begin{array}{l}f(x)=15-x\\ Degree=1,linear\\ \\ f(x)=-3(4x-1-x)\Rightarrow -12x+3+3x=-9x+3\\ Degree=1,linear\\ \\ f(x)=\sqrt[4]{16}+2x\\ Degree=1,linear\end{array}$

The remaining functions that are neither linear nor exponential then we write it in neither

  $\begin{array}{l}f(x)=6x^2+9\\ f(x)=x(18-x)\end{array}$

Conclusion: we get the table as

Exponential	Linear	Neither
$\begin{array}{}$ f(x)=-2{(8)}^x->Growth$$ f(x)=\frac{1}{2}{(3)}^x->Growth$$ f(x)=4{(1.6)}^{\frac{x}{10}}->Growth$$ f(x)=3{(\frac{1}{6})}^x->Decay$\end{array}$	$\begin{array}{l}f(x)=15-x\\ f(x)=-3(4x-1-x)\\ f(x)=\sqrt[4]{16}+2x\end{array}$	$\begin{array}{l}f(x)=6x^2+9\\ f(x)=x(18-x)\end{array}$