Determine if morphism, find kernel and image

[polarbears]

Homework Statement

Determine if the following is a group morphism. Find the kernel and the image if so.
$$f:C_{2} \times C_{3} \rightarrow S_{3}$$ where $$f(h^{r},k^{s})=(1,2)^{r} \circ (123)^{s}$$

Homework Equations

The Attempt at a Solution

I'm stuck on the morphism part. So I know I need to show that $$f(h^{r+x},k^{s+y})=(1,2)^{r+x} \circ (123)^{s+y} = (12)^{r} \circ (123)^{s} \circ (12)^{x} \circ (123)^{y}$$
but I'm not sure how to do that.
Also could someone check my set up?

 

[polarbears]

Been awhile and no response