Define $f: C \rightarrow \mathrm{R}$ by $f(z)=|z|, \forall z \in C$. Then, which of the following is false ?

Define $f: C \rightarrow \mathrm{R}$ by $f(z)=|z|, \forall z \in C$. Then, which of the following is false ?
  1. $f(-z)=f(z), \forall z \in C$
  2. $f(\bar{z})=f(z), \forall z \in C$
  3. $f\left(z^2\right)=(f(z))^2, \forall z \in C$
  4. $f\left(z_1^2+z_2^2\right)=f\left(z_1^2\right)+f\left(z_2^2\right), \forall z_1, z_2 \in C$

Solution

Given, $f(z)=|z|$ (a) $f(-z)=f(z)$ is true $ |z|=|-z| $ (b) $ f(\bar{z})=f(z) \text { is true } $ $ |\bar{z}|=|z| $ (c) $ \begin{aligned} & f\left(z^2\right)=[f(z)]^2 \text { is true } \\ & \left|z^n\right|=|z|^n \\ & \therefore f\left(z^2\right)=\left|z^2\right|=|z|^2=[f(z)]^2 \end{aligned} $ $ \text { (d) } \begin{aligned} & f\left(z_1^2+z_2^2\right)=f\left(z_1^2\right)+f\left(z_2^2\right) \\ \Rightarrow & f\left(z_1^2+z_2^2\right)=\left|z_1^2+z_2^2\right| \\ \Rightarrow & f\left(z_1^2\right)+f\left(z_2^2\right)=\left|z_1^2\right|+\left|z_2\right|^2 \\ \Rightarrow & \left|z_1^2+z_2^2\right| \neq\left|z_1\right|^2+\left|z_2\right|^2 \end{aligned} $ $\therefore$ It is false

Asked in: AP EAMCET 2021 (25 Aug Shift 1)

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