Given $f(A) = \{ac \mid abc \in A \ and \ |a| = |b| = |c|\}$, why is $f(L)$ not regular if $L$ is a regular language?

Question

Given the function:

$$f(A) = \{ac \mid abc \in A \ and \ |a| = |b| = |c|\}$$

Why is $f(L)$ not always regular if $L$ is a regular language?

Is pumping lemma useful in this situation?

Answer 1

Sorry, but I will use different notation from yours.

Hint. Let $A = \{a,b,c\}$ be the alphabet and let $L = a^*cb^*$. Compute the language $f(L) \cap a^*b^*$ and show that it is not regular.