You're passing backroom by value. You need to alter menu()
to take in either an inventory reference or an inventory pointer, preferably the former. This is because when you pass a class object by value, the entire object is copied to a new instance local to the function being called and any changes made to that instance will not effect the original copy.
This should work:
void menu(int zed, inventory& a) {
int z = zed; //you don't need this by the way
int c = 0;
switch (z){//start switch
case 1:
cout << "Enter the amount of Gas you want to change: ";
cin >> c;
a.changegas(c);
break;
case 2:
cout << "Enter the amount of Tires you want to change: ";
cin >> c;
a.changetires(c);
break;
case 3:
cout << "Enter the amount of Soda you want to change: ";
cin >> c;
a.changesoda(c);
break;
case 4:
cout << "Enter the amount of Snacks you want to change: ";
cin >> c;
a.changesnacks(c);
break;
}
}