You can rewrite h(x) so that it doesn't contain g.

When sketching h(x) it is similar to your graph of g(x) but just subtracting x, redraw your sketch of g(x) but subtracting larger values of x as you move long the real line.

The y intercept occurs at x=0, your intercept for g(x) is incorrect.

The x intercept occurs at y=0, the x intercept might not exist, you can tell from your sketches.