Answer
$g(x)$ is linear, $f(x)$ is not
$g(x)=2x-1$
Work Step by Step
For linear functions,
a change of $\Delta x$ units in results in a change of $\Delta y=m\Delta x$ units in $y$.
Using m$=\displaystyle \frac{\Delta y}{\Delta x} =\displaystyle \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$, on the points defined by the first two columns
for f(x), m should be $\displaystyle \frac{3-0}{3-0}=1,$that is,
for a change in x of $\Delta x=+1$, the change in y should be $\Delta y=+1.$
this fails on the next column: x changes from 3 to 6 ($\Delta x=$3)
but the change in y is $5-3=2$,
..so f is NOT linear.
Checking g(x), the first two columns suggest that $m=\displaystyle \frac{5-(-1)}{3-0}=\frac{6}{3}=2$
So, $\Delta y=2\cdot\Delta x:$
x changes from 3 to 6, $\Delta x=3$, $\Delta y=11-5=6$, OK
x changes from $6$ to $10$, $\Delta x=4$, $\Delta y=19-11=8$, OK
x changes from $10$ to $15$, $\Delta x=5$, $\Delta y=29-19=10$, OK
so, g is linear $(g(x)=mx+b$) and has slope 2.
The table shows
$b=g(0)=-1$ so
$g(x)=mx+b$
$g(x)=2x-1$
