Compositions Of 2 Relations, one is the Entire Plane

[teroenza]

Homework Statement

Sketch the compositions

S of R

R of S

Homework Equations

The relations to be... "composition'ed?".
R= R^2, the XY plane, i.e. {(x,y)}, the set pf all (x,y)

S= {(x,y) : y=x^3}, set of all (x,y) such that y=x^3

For example

R of S = {(a,c) : $\exists$b such that a^3=b and {(b,c)}}

The Attempt at a Solution

I feel it should end up being all of R^2, but this seems too simple. For R of S, S maps a into$\sqrt[3]{b}$, then R maps b into, c. But that c is all real numbers. I have done these before, and the teacher wants a graph of a as x, and c as y. Because c will be all real for each b, this will be a succession of vertical lines. But I can see no restrictions on a to prevent the final graph from just being all of the plane (the ac place now).

 

[lippyka]

For S of R, it's more complicated, I think. S maps (a,b) into (\sqrt[3]{a},b). Then R maps that into c. I can't see how to restrict a to prevent the final graph from just being all of the plane (the bc place now).I feel like I'm missing something. Please help.