finding the range of a radical function.

Question

Consider the function f(x) = $\ \sqrt x + \frac {1}{\sqrt x} $

a) Determine domain, range, and roots of $\ f $. Give reasoning

I could figure out the domain easily as what's inside the radical should be greater than or equal to zero. $\ x \ge 0$ so the domain will be

$\ \{x:x \in R, x\ge 0\} $

any ideas about the range and the roots of this?

Answer 1

As $\sqrt x\ge0$ and $x$ is non zero and finite

WLOG $\sqrt x=y^2,y\ne0$

$y^2+\dfrac1{y^2}=(y-1/y)^2+2\ge?$

Alternatively let $u=y^2+\dfrac1{y^2}$

Rearrange to form a quadratic equation in $y^2$

As $y^2$ is real and $y^2>0$ use the discriminant $\ge0$