extension of a bounded linear functional on C*-algebras

Question

Let $A$ be a C*-subalgebra of $B$. Let $f$ be a bounded linear functional on $A$ and assume $g$ be an extension of $f$ on $B$ (may be $||g||\geq ||f||$).

Q. True or false? $|f|\leq |g|_{A}$ ($|f|$ is the absolute value of $f$).