Is the set of points which are the image of all the points in {-1} under f.

As {-1} contains but one point f({-1}) = {f(-1)} = {0}.

Is the set of all points which f maps into the closed interval [-1,1], or allii) f^-1 [[-1, 1]]

points x such that |f(x)|<=1. As the domain of f isRand f(x)>0 for

all x inRthis is the set of all x inRsuch that:

(x+1)^2 <=1

or:

-1<x+1<=1,

or:

-2<=x<=0

which is the closed interval [-2,0].

The use of brackets here is a bit confusing, but I will assume this means:iii) f[ [-1, -1] ] union [1, 3] ]

f([-1,-1] Union [1,3]) = {f(-1)} Union f([1,3]) = {0} Union [1,16]

As we are working iniv) f[f^-1 [ [-3, -1] ] ]Rf^-1([-3,-1]) = NullSet (because there is no real

x such that (x+1)^2 is negative), and f(NullSet)=NullSet, so:

f(f^-1 ([-3, -1])) = NullSet.

RonL