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Subtract transfer functions by subtracting their numerators with a common denominator.

Transfer functions are added by summing their numerators with a common denominator.

Add the transfer functions of the individual systems for a parallel connection.

Multiply the transfer functions of the individual systems for a series connection.

Multiply transfer functions by multiplying their numerators and denominators respectively.

Divide transfer functions by flipping the second transfer function and then multiplying.

Break down complex transfer function diagrams into simpler series and parallel combinations.

Common poles and zeros can be canceled in the numerator and denominator of the transfer function.

Time delays can be incorporated into a transfer function using $e^{-sT_d}$ where $T_d$ is the delay time.