Well, there is not just one function that g(x) could be.
But it's easy to find such a function.  We could use the 
standard formula for a parabola,
  
y = a(x-h)²+k with vertex at (h,k) 

to find a function with a vertex at (1,-2).  To make it
easy, we choose "a" as 1, positive so it will open 
upward making the vertex a minimum point.

g(x) = (x-1)²-2

Here's the graph of g(x):



Then 

h(x) = -3g(x)-1

h(x) = -3[(x-1)²-2]-1
h(x) = -3(x-1)²+6-1
h(x) = -3(x-1)²+5

Compare that to y = a(x-h)+k  with vertex (h,k)

and we see this parabola h(x) has vertex (1,5), which is 
a maximum point because "a" is negative making the 
parabola open downward:  Here are both graphed together.
The red graph is g(x) and the green one is h(x).



Edwin