(1) g(x)=-2×2ˣ⁻¹+1, -(2¹⁺ˣ⁻¹)+1=-2ˣ+1.
(2) f(x)=-x²-2x+3 is blue and g(x) is red.
The x-intercepts are the roots of f(x) and the solution to -x²-2x+3=0=-(x+3)(x-1).
(3) Axis of symmetry for f is x=-1 (green line). Asymptote passes vertically through the vertex of the parabola.
(4) Range of f is (-∞,4].
(5) Range of g is (-∞,1).
(6) I guess you mean f(0)-g(0)=3-0=3.
(7) Imagine pulling down the blue curve so that it’s just below the x-axis (so there can be no x-intercepts and hence no real solution). The “hump” or vertex must be below the x-axis which means the y intercept at 3 must be lower than y=-1. The constant 3 is the current y-intercept and it needs to move more than 4 units down so that the vertex is below the x-axis, so k<3-4, k<-1.