Question

$f\left(x\right)=\log x^2$and $\varphi \left(x\right)={\left(\sqrt{x}\right)}^2.$
Are both functions equal?

Answer 1

Given $f\left(x\right)=\log x^2$

For logarithm to be defined, $x^2>0\Rightarrow x\in R$

$D\left(f\right)=R$

$R\left(f\right)=R$

$\varphi \left(x\right)=(\sqrt{x}{)}^2$

Square root function is defined for all $x\ge 0$

$D\left(\varphi \right)=[0,\infty )$

$R\left(\varphi \right)=R$

Since, $D\left(f\right)\ne D\left(\varphi \right)$

Hence, the two functions are not equal