Let $g:\mathbb{R} \to \mathbb{R}$ a continuous function such that exists $p \geq 1$ and an constant $C>0$ that $$|g(t)| \leq C(|t|+|t|^{p})\,\,\,\forall t \in \mathbb{R}$$
Let be $\eta(t)$ be an function in $C_0^{\infty}(\mathbb{R})$ and define
$g_1(t)=\eta(t)(t+g(t))$ and $g_2(t)=(1-\eta(t))(t+g(t))$
My question is: It is possible to construct $\eta(t) \in C_0^{\infty}(\mathbb{R})$ such that
$|g_1(t)| \leq C \min\{1,|t|\}$ and $|g_2(t)| \leq C|t|^{p}$ ?
The possibility of doing the above construction will help me in the process of regularization of an Elliptic PDE, because if the above is true I can divide a harder problem in two easier problems. Thank you very much for the help, any hint will be very helpful.