Allow a custom path in Firebase functions custom domain

.changeset/dull-nails-kneel.md

@@ -0,0 +1,5 @@
+---
+'@firebase/functions': patch
+---
+
+Allow a custom path in Firebase functions custom domain

packages/functions/src/service.test.ts

@@ -70,6 +70,11 @@ describe('Firebase Functions > Service', () => {
       assert.equal(service._url('foo'), 'https://mydomain.com/foo');
     });
 
+    it('correctly sets custom domain with path', () => {
+      service = createTestService(app, 'https://mydomain.com/functions');
+      assert.equal(service._url('foo'), 'https://mydomain.com/functions/foo');
+    });
+
     it('prefers emulator to custom domain', () => {
       const service = createTestService(app, 'https://mydomain.com');
       connectFunctionsEmulator(service, 'localhost', 5005);

packages/functions/src/service.ts

@@ -122,7 +122,7 @@ export class FunctionsService implements _FirebaseService {
     // Resolve the region or custom domain overload by attempting to parse it.
     try {
       const url = new URL(regionOrCustomDomain);
-      this.customDomain = url.origin;
+      this.customDomain = url.origin + url.pathname;
       this.region = DEFAULT_REGION;
     } catch (e) {
       this.customDomain = null;