Answer
f(x) is exponential, g(x) is not.
$f(x)=2\cdot 2^{x}$
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)}$:
$\displaystyle \frac{1}{1/2}=2,\quad\frac{2}{1}=2,\quad\frac{4}{2}=2,\quad\frac{8}{4}=2$
so it is exponential, $f(x)=a\cdot 2^{x}.$
Since $f(0)=2$, then $a=2.$
Thus, $f(x)=2\cdot 2^{x}$
For the values of g(x),
$\displaystyle \frac{0}{3}=0,\quad\frac{-1}{0}=$undefined
so it is not exponential,