A question about functions in the first-order language

Question

Let $L$ be a first-order theory and a set $T$ be the domain of discourse of $L$. Also, let $F$ be a $n$-ary function of $L$. Then, must the domain of $F$ be $R^n$? If so, it seems that we can't have diverse functions in algebra. However,if the $F$'s domain hasn't to be $R^n$, it also quite strange that the rule $\forall x: α\rightarrow α(x/t)$ will sometimes go wrong. For example, we can obtain $lg(x)+lg(-x)=lg(-x)+lg(x)$ from $\forall x \forall y(x+y=y+x)$.