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Determine the domain of the function $f(x)=\left | x \right |+1$

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This would help you. Domain(f) =$(-\infty,\infty)$ and Range(f)= $[1,\infty).$ To draw graph, you just have to shift graph of $|x|$ one unit upward because we can tranform $f(x)$ to $f(x)+a$ by shifting $a$ unit(s) upward for $a>0.$

The domain of the function f(x) = |x|+1  is the real line itself.