Let $f$ be a functions from $\mathbb{R^+}$ to $\mathbb{R^+}$ defined by $f(x)=x^x$
Determine whether the function $f$ injective or surjective ?
I saw here similar questions regarding this function but this differ from other problem, I will tell you what I tried
Attempt:
$\underline{Surjective}$
for $f(x) \in \mathbb{R^+}, $ no element $x \in \mathbb{R^+} $ such that $f(x)=0 \Rightarrow x^x=0$
Thus $f$ is not onto
I have no idea how to prove it or disprove it as a surjective,
Actually this problem is very strange to me, I referred this problem How can we describe the graph of $x^x$ for negative values? then I thought my problem would be wrong,
Any help would be greatly appreciated regarding this problem.
Best Answer
One can see that it's not injective since $f(1/2)=(1/2)^{1/2}=1/\sqrt{2}=(1/4)^{1/4}=f(1/4)$