Question 15.22: The voltage transfer function for a certain lowpass circuit ...

The squared voltage gain is given by

A_ν^2=\frac{1}{1+\left(f / f_0\right)^2}

The squared reference  ( dc )  gain is A_0{ }^2=A_ν{ }^2(0)=1. The half-power bandwidth consists of the frequencies for which

\frac{A^2}{A_0^2} \geq \frac{1}{2} \Rightarrow \frac{1}{1+\left(f / f_0\right)^2} \geq \frac{1}{2} \Rightarrow \frac{f}{f_0} \leq 1 \Rightarrow f \leq f_0

The half-power bandwidth is f_0 .