The relation between $P(\mathbf{X}\geq t)$ and $P(\Phi(\mathbf{X}) \geq \Phi(t))$ for any non-decreasing non-negative function $\Phi$, non-negative $t$ and for positive random variables $\mathbf{X}$ is such that
$$ P(\Phi(\mathbf{X}) \geq \Phi(t)) \geq P(\mathbf{X}\geq t) $$
For functions like $\Phi(x) = x^2$, it can be easily imagine such an inequality as the 'size' of the sample space increases. For example if $x \in (0,2)$ then $x^2 \in (0,4)$. But what happens in case of other functions like $\Phi(x) = \sqrt{x}$ ?. Here 'size' of the sample space decreases, if $x \in (0,4)$, then $\sqrt{x} \in (0,2)$ only. In such situation how one can show that above inequality holds ?