Answer
$\mathrm{Even}.$
Work Step by Step
$\mathrm{Function\:Parity\:Definition:} $
$\mathrm{Even\:Function:}\:\: $ A function is even if $\ h(-t)=h(t)\ $ for all $\ x\in \mathbb{R}. $
$\mathrm{Odd\:Function:}\:\: $ A function is odd if $\ h(-t)=-h(t)\ $ for all $\ x\in \mathbb{R}. $
$h(t)=2|t|+1$
$h(-t)=2|(-t)|+1$
By applying the absolute rule $\ |-a|\:=\:|a|\:\ $, we have:
$h(-t)=2|t|+1$
Now,
$-h(t)=-(2|t|+1)$
$-h(t)=-2|t|-1$
Since,
$h(-t) = h(t)\mathrm{,\:therefore\:}2|t|+1\mathrm{\:is\:an\:even\:function}$
$h(-t)\ne -h(t)\mathrm{,\:therefore\:}2|t|+1\mathrm{\:is\:not\:an\:odd\:function}$
