Given the functions:
f(n) = n^2 +n
We need to find the function g(n) + f(n).
==> f(n)+ g(n)= (f+g)(n) = 2x-1 + n^2 + n = n^2 + 3n -1
==> (f+g)(n)= n^2 + 3n -1
The domain = R ( all real numbers)
To find the range, we need to determine the maximum or minimum value of the function.
Since the coefficient of n^2 is positive, then the parabola facing upward. Then, the function has minimum value at the vertex.
Now we will need to find the vertex.
vx = -b/2a = -3/2
vy= (f+g)(vx)= (f+g)(-3/2)= (-3/2)^2 + 3(-3/2) -1 = 9/4+9/2 -1= (9-18 - 4)/ 4 = -13/4 = - 3.25
==> Then, the vertex is the point ( -3/2, -3.25)
Now since the parabola is facing upward, then the range is all y-values such that y >= -3.25
Then, the range is y >= -3.25
See the graph below for further explanation.