Functions f and g are defined by
f(x)= 2x-2, if x<1 and
f(x)= x^2 -1, if x ≥1
and g(x) = |x|-x
Show that the composite function g。f exists for all values of x.
I have totally no ideas on showing this, can anyone help me?
If f : A -> B and g : B -> C for some sets A, B, C, then the composition g o f : A -> C exists regardless of the particular definitions of f and g. The definition of the composition proves its existence; there is no issue here that requires proving that composition is well-defined.
What emakarov said is of course true. Though you could consider the 2 cases.
When x<1, g(f(x))=g(2x-2)= abs(2x-2)-2x+2 =2-2x-2x+2=4(1-x)
when x>_1, g(f(x))= g((x^2-1)=abs(x^2-1)+1-x^2=x^2-x^2-1+1=0
so in both cases the composite function is well defined.