Square Root in Denominator

[harpazo]

Given f(x) = 5/sqrt{x^2 + 1}, the domain is all real numbers but I do not know the reason why that is the case.

For rational functions, we set the denominator greater than or equal to 0 and solve for the variable. In this case, the radicand is set this way:

x^2 + 1 > 0 NOT greater than or equal to 0.

Why?

 

[Harry_the_cat]

x^2 is greater than or equal to 0. That is, the smallest it can be is 0. When we add 1, to get x^2 + 1, the smallest it can be is 1, so must be greater than 0.

 

[Dr.Peterson]

Given f(x) = 5/sqrt{x^2 + 1}, the domain is all real numbers but I do not know the reason why that is the case.

For rational functions, we set the denominator greater than or equal to 0 and solve for the variable. In this case, the radicand is set this way:

x^2 + 1 > 0 NOT greater than or equal to 0.

Why?

I think you meant to say the right thing, which is "we set the radicand greater than or equal to 0". Your question is, why don't we say x^2 + 1 ≥ 0 in this case, as we do when the radical is in the numerator. But you aren't talking about rational functions at all, but about functions containing a radical.

The answer is that there are two things that can go wrong here: you might be taking the square root of a negative number (which is why the radicand must not be negative), and you might be dividing by zero (which is why the radicand must also not be zero). The combination of these two facts leads us to say it must be positive, when the radical is in the denominator.

 

[harpazo]

I think you meant to say the right thing, which is "we set the radicand greater than or equal to 0". Your question is, why don't we say x^2 + 1 ≥ 0 in this case, as we do when the radical is in the numerator. But you aren't talking about rational functions at all, but about functions containing a radical.

The answer is that there are two things that can go wrong here: you might be taking the square root of a negative number (which is why the radicand must not be negative), and you might be dividing by zero (which is why the radicand must also not be zero). The combination of these two facts leads us to say it must be positive, when the radical is in the denominator.

A perfect reply.