if $ b(x) <f(x) \leq g(x)$ where $b, g \in \Theta(x)$ can we state $f(x) \in \Theta(x)?$ I restate the question:
if $ b(x) &lt;f(x) \leq g(x)$ where $b, g \in \Theta(x)$ can we state $f(x) \in \Theta(x)?$
notice that $b(x)&lt; f(x)$ and not $b(x) \leq f(x)$.
In other words what if

$f(x) \in \omega(b(x)) $ and

$f(x) \in O(g(x))$

can we state $f(x) \in \Theta(x)$?
b(x) is between $c_1·x$ and $c_2·x$ if x is large. g (x) is between $d_1·x$ and $d_2·x$ if x is large. So f (x) is between $c_1·x$ and $d_2·x$ if x is large, which makes it $\Theta(x)$.