First, let's use the values on the table to find the linear function f(x).
A linear function passing through points (a, g(a)) and (b, g(b)) using the following equation:
[tex]g(x)-g(a)=\frac{g(b)-g(a)}{b-a}(x-a)[/tex]Since the line describing g(x) passes through points (-6, 14) and (-3, 8), we have:
a = -6
g(a) = 14
b = -3
g(b) = 8
So, g(x) is given by:
[tex]\begin{gathered} g(x)-14=\frac{8-14}{-3-(-6)}(x-(-6)) \\ \\ g(x)-14=\frac{-6}{-3+6}(x+6) \\ \\ g(x)-14=-2(x+6)_{} \\ \\ g(x)=-2x-12+14 \\ \\ g(x)=-2x+2 \end{gathered}[/tex]Now, we need to compute (f+g)(x) to find its range.
We have:
[tex](f+g)(x)=f(x)+g(x)=x^{2}+2x-6-2x+2=x^{2}-4[/tex]Since the roots of (f+g)(x) are -2 and 2, the vertice of the parabola has x-coordinate 0 (the middle between the roots). At x = 0, we have:
[tex](f+g)(x)=0^{2}-4=-4[/tex]Also, since the coefficient of x² is 1, which is positive, this function represents a parabola opened upwards.
So, the minimum value of that function is y = -4. So, the range of the function is all real values equal to or greater than -4.
In interval notation, the range is written as
[tex](-4,\infty)[/tex]