if g(n) =2n-1 and f(n)= n^2  + n then what is g(n) + f(n)? also, state the domain and range of the resulting function.Thanks

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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Given the functions:

g(n)= 2n-1

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.

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