Condition on $f$ that will imply that $\operatorname{supp}(f*g) = \operatorname{supp}(f)$

Question

$\DeclareMathOperator{\supp}{supp}$ Given $h,k \in L_1(\Bbb R)$ ,

define $(h*k)(x) = \int_{\Bbb R} h(t)k(x-t)dt$.

for any function $h$, define $\;\supp(h) = \{x:h(x)\ne 0\}$.

Now ,let $f,g \in L_1(\Bbb R)$.

I have showed that $\supp(f*g)\subset \supp(f)+\supp(g)$.

Now I want to give a condition for a function $f\ne 0$ that will imply that $\supp(f*g) = \supp(f)$.

So I want a condition that will reduce to $f(x) \ne 0 \iff \int f(t)g(x-t)dt \ne0 $.

I'm not sure what condition will imply that.

Thanks for helping.