# A question regarding real valued function

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.

• 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 Oct 22, 2014 at 17:20
• The question being given to me is $\sqrt{x-1}$. Oct 22, 2014 at 17:21
• Each of the functions are defined for some range. What do you want to ask actually in the question?
– user170039
Oct 22, 2014 at 17:22
• 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... Oct 22, 2014 at 17:23
• 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 $\mathbb{R}$. Oct 22, 2014 at 17:29

After examining all formulas, I got the following diagram:

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