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Funtion Definitions *Need editor hold on*

Mathematics
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If you define the folowing functions as such: \[f:[0,\infty )\rightarrow [0,\infty )\quad f\left( x \right) :=\sqrt { x } \\ g:[0,\infty )\rightarrow { R }\quad g\left( x \right) :=\sqrt { x }\] Is the composite \( f\circ g\) properly defined? I would think it is NOT as g(x) maps to all Real numbers, while the domain of f(x) is \([0,\infty)\).
R is codomain for the second function it maps to the Range which is [0,infinity)
its defined just fine

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Other answers:

Ah, codomain not the range.
\[\sqrt{x}\ge0\] so \[\sqrt{\sqrt{x}}\] is fine
@zzr0ck3r Thanks, codomain always confuses me. But I got it now.
codomain is what you are allowed to map to, range is what you map to

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