  Bent functions as optimal combinatorial objects are difficult to characterize
and construct. In the literature, bent idempotents are a special class of bent
functions and few constructions have been presented, which are restricted by
the degree of finite fields and have algebraic degree no more than 4. In this
paper, several new infinite families of bent functions are obtained by adding
the the algebraic combination of linear functions to some known bent functions
and their duals are calculated. These bent functions contain some previous work
on infinite families of bent functions by Mesnager \cite{M2014} and Xu et al.
\cite{XCX2015}. Further, infinite families of bent idempotents of any possible
algebraic degree are constructed from any quadratic bent idempotent. To our
knowledge, it is the first univariate representation construction of infinite
families of bent idempotents over $\mathbb{F}_{2^{2m}}$ of algebraic degree
between 2 and $m$, which solves the open problem on bent idempotents proposed
by Carlet \cite{C2014}. And an infinite family of anti-self-dual bent functions
are obtained. The sum of three anti-self-dual bent functions in such a family
is also anti-self-dual bent and belongs to this family. This solves the open
problem proposed by Mesnager \cite{M2014}.
