$\huge \begin{align} f' &= f\left( \frac{c-v_{\text{target}}}{c} \right) \\ f'' &= f'\left( \frac{c}{c+v_{\text{target}}} \right) \\ f''&= f\pa{\frac{c-v_{\text{target}}}{c}}\pa{\frac{c}{c+v_{\text{target}}}} \\ f'' &= f\left( \frac{c-v_{\text{target}}}{c+v_{\text{target}}} \right)\\ f' &= f\left( \frac{c \pm v_{\text{obs}}}{c \mp v_{\text{src}}} \right) \end{align} $