Question

Determine the functions f: R -> R are onto, f(x) = |x| + x

Answer 1

$$x<0: f(x)=-x+x=0$$

$$x\ge0: f(x)=x+x=2x$$

Then

$$f(x)= \begin{cases} 0& x<0 \\ 2x & x\ge0 \end{cases}$$

Domain: $(-\infin, \infin)$

Range: $[0, \infin)$

The function $f: R\to R$ is not onto because there is no $x\in \R$ with $|x|+x=-1,$ for instance.