These days, the most common use for full-width images I see are hero images.
They take up the full width of the viewport, and their height is maintained in relation to the width. That is, if the image starts out at 2000 x 1000, then when it is resized to 1000 width, the height will be 500.
Aside from hero images, I don't see a lot of images being used as backgrounds. This might be because there isn't a simple solution for having an image be full width and full height.
After all, mobile viewports may not be very tall, but vertical scrolling is much more common than horizontal scrolling, which means that the height of a website is effectively infinite.
One solution might be to have a fixed background image, so that it essentially moves with the viewport as it scrolls - this might be your plan already.
For smaller width screens, I think the best you can hope for is an image that keeps its meaning as the user only sees the middle of it. This lets you keep the full height of the image, but you crop the sides.
A final solution is to find a background colour to sit behind the image that complements it, but there'll always be an "edge" between the image and the colour.