Dolly Robinson

2021-08-10

A table for a function f is given to the right.
​(a) Determine whether function f represents exponential​ growth, exponential​ decay, or linear growth.
​(b) Find a formula for f.

yunitsiL

$\begin{array}{}2& -1& 0& 1& 2\\ 1.28& 3.2& 8& 20& 50\end{array}$
a) $f\left(x\right)$ represents exponential growth (decay) according as $\frac{f\left(x+1\right)}{f\left(x\right)}=r>1\left(<1\right)$.
It represents linear growth if $f\left(x+1\right)-f\left(x\right)=$ constant $\mathrm{\forall }x$
Now $\frac{f\left(-1\right)}{f\left(-2\right)}=\frac{3\cdot 2}{1\cdot 28}=\frac{5}{2}.$
Also $\frac{f\left(0\right)}{f\left(-1\right)}=\frac{f\left(1\right)}{f\left(0\right)}=\frac{f\left(2\right)}{f\left(1\right)}=\frac{5}{2}.$
b) Since the ratio is a constant with r > 1 , f represents exponential growth.
Let $f\left(x\right)=A\cdot {\left(\frac{5}{2}\right)}^{x}$
$f\left(0\right)=A=8$
Hence $f\left(x\right)=8\cdot {\left(\frac{5}{2}\right)}^{x}$

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