Chapter 4 - Exponential Functions - 4.2 Comparing Exponential and Linear Functions - Exercises and Problems for Section 4.2 - Exercises and Problems - Page 154: 8

A) Exponential B) $h(x)=14(0.9)^x$

Given $$ \begin{array}{c|c} \hline x & h(x) \\ \hline 0 & 14 \\ 1 & 12.6 \\ 2 & 11.34 \\ 3 & 10.206 \\ 4 & 9.185 \\ \hline \end{array} $$ A) We know that a function is linear if the differences of successive values of the function is a constant. A function is exponential if the ratios of successive values of the function is a constant. $$ \begin{aligned} &h(1)-h(0)=12.6-14=-1.4\\ &h(2)-h(1)=11.34-12.6=-1.26 \end{aligned} $$ Thus, the function is not linear. The ratios gives $$ \begin{aligned} & \frac{h(1)}{h(0)}=0.9 \\ & \frac{h(2)}{h(1)}=\frac{11.34}{12.6}=0.9 . \end{aligned} $$ Hence, the function is exponential. B) We know that $$ h(x)=ab^x $$ Thus, $$ \begin{aligned} & 14=h(0) \\ & 14=a b^0 \\ & 14=a . \end{aligned} $$ $$ \begin{aligned} & 12.6=h(1) \\ & 12.6=14 b\\ &b=\frac{12.6}{14}=0.9 \end{aligned} $$ Hence: $$ h(x)=14(0.9)^x $$