Is a circuit a bandpass filter only when it's a high pass filter followed by a low pass filter? Or can it ever be a low pass filter and then a high pass filter (cascaded together)?

Does the order matter if it's then followed by a gain/amplifier stage?

Even if an amplifier circuit does not have band-pass filters, there is a limit on the band of frequencies that it can pass. The lowest frequency is determined by the value of the coupling capacitors. The highest frequency is limited by any capacitance in parallel with the input and output.

To pass a pre-determined band of frequencies and block all others, high-pass and low-pass filters may be used.

If this band of frequencies is followed by an amplifier, it must be designed to pass the entire band of frequencies. The order of the frequencies doesn't matter.

A circuit can indeed function as a bandpass filter when it consists of a sequence of a high pass filter followed by a low pass filter (also known as cascading filters). This arrangement allows only a specific range of frequencies to pass through while attenuating frequencies outside that range.

On the other hand, if a circuit is configured as a low pass filter followed by a high pass filter, it will not function as a bandpass filter. This arrangement will result in frequencies that are first attenuated by the low pass filter, and then any remaining frequencies will be passed through by the high pass filter. It effectively behaves as a combination of a low pass and a high pass filter, but not a bandpass filter.

Now, let's consider the inclusion of a gain/amplifier stage after the filter configuration. The order in which the gain stage is placed in relation to the filters can affect the overall response of the circuit. If the gain stage is placed before the filters, it will amplify all frequencies equally, including those outside the desired bandpass range. However, if the gain stage is placed after the filters, it will only amplify the frequencies that managed to pass through the filters. This can significantly affect the gain characteristics of the circuit.

In summary, a circuit can operate as a bandpass filter when it consists of a high pass filter followed by a low pass filter, and the order does matter when it is followed by a gain/amplifier stage.