$f(x)=x+2+2\ln x.$. Find the number of solutions of $f.$ Help?

Question

I have found that f is increasing, its domain is Df=(0,+oo) and f((0,+oo)) = R How do i find the number of solutions of f and how am i explaining it? Thanks.

Answer 1

For $x>0$,

$$f'(x)=1+\frac 2x=\frac {x+2}{x} >0$$

$$\implies $$ $f $ is strictly increasing from $(0,+\infty) $ to $\mathbb R $

$f $ is continuous at $(0,+\infty) $ thus it is bijective.

the equation $f (x)=0$ has only one root.