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I have a question regarding real-valued function:

Which of the following cannot possibly be the rule of any real-valued function?

A) $y=\sqrt{x-1}$

B) $y=\sqrt{x-1}+\sqrt[3]{2+x}$

C) $y=\sqrt{x-1}+\sqrt{2+x}$

D) $y=\sqrt[3]{x-1}+\sqrt{2+x}$

E) $y=\sqrt[3]{x-1}+\sqrt[3]{2+x}$

I have this feeling that if A is not a real valued function, then B and C too are not real valued functions. Since $\sqrt{x-1}$ is not much different from $\sqrt{x+2}$, then D is also not a real valued function.

Can someone assist?

Thank you.

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  • $\begingroup$ Are you sure there's no $\sqrt{1-x}$ or the like instead? Whether we see a real-valued function before us largely depends on the domain we want to define it on $\endgroup$ Commented Oct 22, 2014 at 17:20
  • $\begingroup$ The question being given to me is $\sqrt{x-1}$. $\endgroup$
    – Novice
    Commented Oct 22, 2014 at 17:21
  • $\begingroup$ Each of the functions are defined for some range. What do you want to ask actually in the question? $\endgroup$
    – user170039
    Commented Oct 22, 2014 at 17:22
  • $\begingroup$ This is what I have thought as well. Apparently, my teacher does not think so... There maybe a possibility that there are typos in the question, or I might have missed something crucial... $\endgroup$
    – Novice
    Commented Oct 22, 2014 at 17:23
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    $\begingroup$ I think that the question is: "Which of the following possibly be the rule of any real-valued function?" int this case is (E), the domain is $\mathbb{R}$. $\endgroup$ Commented Oct 22, 2014 at 17:29

1 Answer 1

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After examining all formulas, I got the following diagram: enter image description here

With the following formulas:

$\color{purple}{y =\sqrt{x-1}}$

$\color{red}{y = \sqrt{x-1}+\sqrt[3]{2+x}}$

$\color{blue}{y = \sqrt{x-1}+\sqrt{2+x}}$

$\color{green}{y = \sqrt[3]{x-1}+\sqrt{2+x}}$

$\color{orange}{y = \sqrt[3]{x-1}+\sqrt[3]{2+x}}$

It shows that all formulas have real values, so the answer is none. Perhaps the question was "Which of the following is the function of any real-valued function?". In that case, the answer is E. While all the other function stop at a moment, the orange function doesn't have an end, and has a real value for all $x$'s

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