Maximizing Range of 0 < x < 1: Comparing Expressions x-1 and x-1/2

[Schnellmann]

Homework Statement

I was doing some multiple choice questions. One question was:

"Which of the expressions below has the largest value for 0 < x < 1

The two relevant options (dismissing those obviously wrong) were:

x-1 (can't do superscript on my iPhone - that should read X to the power of -1 ie 1/x)

Other option was x-1/2 meaning one over the square root of x

I took the question to mean which equation gives the greatest range for values of x between zero and one.

The answer was given as 1/x but that confuses me because a) both have the range from 1 < range < infinity It is true that for any individual x then 1/x is larger than one over the square root of x but I thought that you couldn't compare infinities in that way.

Also I thought that if you take the square root of x then doesn't it have a positive and negative root such that you actually have a range of - infinity < range < -1 as well as the range 1 < range < infinity.

Where am I going wrong?

Homework Equations

The Attempt at a Solution

 

[PeroK]

Homework Statement

I was doing some multiple choice questions. One question was:

"Which of the expressions below has the largest value for 0 < x < 1

The two relevant options (dismissing those obviously wrong) were:

x-1 (can't do superscript on my iPhone - that should read X to the power of -1 ie 1/x)

Other option was x-1/2 meaning one over the square root of x

I took the question to mean which equation gives the greatest range for values of x between zero and one.

The answer was given as 1/x but that confuses me because a) both have the range from 1 < range < infinity It is true that for any individual x then 1/x is larger than one over the square root of x but I thought that you couldn't compare infinities in that way.

Also I thought that if you take the square root of x then doesn't it have a positive and negative root such that you actually have a range of - infinity < range < -1 as well as the range 1 < range < infinity.

Where am I going wrong?

You may have misinterpreted the question. What about this:

For a given $x$ between $0$ and $1$, which of the following expressions gives the largest value?

On a second point about square roots. The square root of $1$ is $1$. As in, $\sqrt{1} = 1^{1/2} = 1$ In other words, the square root has a unique, positive value.

There is another "root", which is: $y = -\sqrt{x}$ which also has the property that $y^2 = x$. But this $y$ is not the square root.