Answer
g(x) is exponential, f(x) is not.
$g(x)=4\displaystyle \cdot(\frac{1}{5})^{x}$
Work Step by Step
If a function x is an exponential function, then it has form
$f(x)=a\cdot b^{x}$
When we observe an increase of x by 1 unit, $f(x+1)=ab^{x+1}.$
what follows is that the ratio $\displaystyle \frac{f(x+1)}{f(x)}=\frac{ab^{x+1}}{ab^{x}}=b.$
(the ratio is constant when arguments increase by 1).
Thus, if the x increases by unit, the function value is multiplied with b.
We use this to recognize whether a table of function values represent an exponential function or not.
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The x-values increase by 1 unit.
For the values of f(x) we observe $ \displaystyle \frac{f(x+1)}{f(x)}$:
$\displaystyle \frac{200}{100}=2,\quad\frac{400}{200}=2,\quad\frac{600}{400}\neq 2,$
so f is not exponential.
For the values of g(x)
$\displaystyle \frac{20}{100}=\frac{1}{5},\quad \displaystyle \frac{4}{20}=\frac{1}{5},\quad \displaystyle \frac{0.8}{4}$=$\displaystyle \frac{1}{5}$,\quad $\displaystyle \frac{0.16}{0.8}$=$\displaystyle \frac{1}{5}, \quad$
so g is exponential, $\mathrm{g}(\mathrm{x})=\mathrm{a}(\mathrm{b}^{\mathrm{x}}).$
Since $g(0)=4$, then $a=4$
Thus, $g(x)=4\displaystyle \cdot(\frac{1}{5})^{x},\qquad ($which can also be written as $4\cdot 5^{-x})$
