There is an error in Q1: "x greater than or equal -5" should be "x greater than or equal 5"; otherwise the piecewise function doesn't make sense. The domain of f(x) is x≥0, because the pieces of the function provide continuity ∀x∈[0,∞).
Q1: f(x)=
{ x2 | 0≤x<2
{ x+2 | 2≤x<5
{ sgn(x) | x≥5
In the graph below there are three functions with unrestricted domains: the blue curve is f(x)=x2; green line is f(x)=x+2; red lines are f(x)=sgn(x). The orange verticals mark the critical points of the piecewise function (x=0, 2 and 5).
The piecewise function f(x) is a combination of the three complete functions shown above.
f(x) starts at (0,0) and traces the blue curve up to where it intersects the green line at (2,4). It then continues along the green line up to (but not including) the orange vertical at x=5 then there is a discontinuity as the function drops to (5,1) on to the red line where it stays forever after.
The range starts at (0,0) making the lower limit of the range=0. The range increases between x=2 and 5. f(5)=1, but just before the point (5,7) f(x) approaches 7, so the range is [0,7).
Q2: f(x)=
{ x-1 | -3≤x<-2
{ x2 | -2≤x<0
{ sgn(x)-1 | 0≤x<8
The blue curve is f(x)=x2; green line is f(x)=x-1; red lines are f(x)=sgn(x)-1. The orange verticals mark the critical points of the piecewise function (x=-3, -2, 0 and 8). The functions shown are components of piecewise f(x).
The domain is [-3,8). Piecewise f(x) starts at (-3,-4) on the green line and continues along this line until just before (-2,-3) when it jumps discontinuously to (-2,4) on to the blue curve. It continues along the blue curve until it meets the red line at (0,0), and continues almost up to x=8. It then follows the red line till just before (8,0). The low limit of piecewise f(x) is (-3,-4), while the high limit is (-2,4). Because of the blue curve f(x) is continuous between x=-2 and x=0; however, the range is not continuous, because it has no values between f(x)=-3 and f(x)<0. The range can be expressed: f(x)∈[-4,-3)⋃[0,4] where ⋃ represents union of the two sets of values in the intervals.