The height of a laser beam trajectory relative to a datum is modeled by the function H(x)H(x)H(x), where
H(x)=2∣3x−9∣+4,x≥0 H(x) = 2|3x - 9| + 4, \quad x \ge 0 H(x)=2∣3x−9∣+4,x≥0The minimum point on the beam's path is at the vertex VVV.
Determine the coordinates of VVV.
Find the values of x x\,x for which H(x)=x+7H(x) = x + 7H(x)=x+7.
Consider the family of linear paths defined by G(x)=kx+1G(x) = kx + 1G(x)=kx+1, where k k\,k is a constant. Given that the equation H(x)=G(x)H(x) = G(x)H(x)=G(x) has exactly two distinct solutions,
find the range of possible values for kkk.
