If a computer with dual hard drives has one broken hard drive will it still work?

If the one with the bootloader and the OS on it is not the one broken, the extra hard drive that is broken will simply not be able to be accessed. An easy way to check if the bootloader and OS are on the same one, is unplug one drive and start the computer.

"dual hard drives" if you mean raid by this (as far as your OS knows both drives are one), then you would have to play with recovery if one of the two drives failed. Raid 0 is stiped, both hard drives spin to write one file (faster), raid 1 means that the two drives mirror each other and if one fails an exact 1:1 copy is still there.

Depends on how the drives were configured, and what was on the drive that failed.

If the drives were configured as RAID 0 (mirrored) the computer will act normally.

If the drives were configured as RAID 1 (striped) the computer will not start if either drive is dead.

If the drives were independent, the computer will only boot if the drive with the operating system is working.

It kind of depends. The hard drive will be just fine, but if it isn't the hard drive that you had the OS installed on, then you will need to reinstall the OS onto the working HDD.

If you are talking about physical hard drives and assuming the bad drive is not the one holding the system files, then it is possible your machine may function; but it will alert you of a faulty condition

Depends. If the dead drive had the OS on it, then no. If the other drive had the OS, or it was in a RAID array with fault tolerance, then yes, it would still work.