firebase/functions does not respect a custom path the in the customDomain parameter

[segevfiner]

Operating System

macOS 14.6.1

Browser Version

Chrome/127.0.6533.120

Firebase SDK Version

10.12.5

Firebase SDK Product:

Functions

Describe your project's tooling

Vue app with Vite

Describe the problem

Trying to set a custom domain using getFunction(app, 'https://example.org/functions') doesn't work due to the SDK not handling it correctly.

I already have a fix. I'm only opening this issue since it is required to do so to open a PR.

Steps and code to reproduce issue

Just do getFunction(app, 'https://example.org/functions'), the functions get's send to just the origin without the path.

[google-oss-bot]

I couldn't figure out how to label this issue, so I've labeled it for a human to triage. Hang tight.