Can someone explain how do you know whether the composition of functions are defined or not?
E.g. and
I haven't done these in ages.
Ran(f) = [-0.25,infinity)
Ran(g) = [0,infinity)
so for f o g to exist, the ran(g) must be a subset of dom(f), which is true.
for g o f to exist, the ran(f) must be a subset of dom(g), which is is false, since the ran(f) is NOT a subset of the dom(g).
If you think carefully about sets it makese sense.
f states that it can map x onto the real numbers as long as it's a real number
(basically why you may see R -> R, that is what the function is mapping the set of x-values to)
g states that it can map x onto the real numbers as long as it's a positive real number or 0.
g can't map f onto the real numbers, since not all values of f are positive.