Could someone explain why f is not a linear operator while g is?
What are the criteria to distinguish if a function is a linear operator?
Is linear transformations another way to call for linear operators? Our book is not clear on this.
Thanks in advance.
The difference between linear operator and linear transformation is vague and depends on your course. Some instructors may want to distinguish these notions, while some won't. One of the common approaches is to say that linear operator is a particular case of linear transformation, when the same vector space is both domain and codomain. You may see this question for more details.
A linear transformation from vector space $V$ into vector space $W$ (over the same field $F$) is a function $T$ from $V$ into $W$ such that $\forall \alpha,\beta\in V$ and $\forall c\in F$: $$T(c\alpha+\beta) = c(T\alpha)+T\beta.$$
This is just a definition that you should be able to find in any linear algebra book.
Apply the definition to check whether $f$ is a linear transformation or not. Let $v_1=(a_1,b_1)$ and $v_2=(a_2,b_2)$, then $$f(v_1+v_2)=(a_1+a_2)\cdot(b_1+b_2)+1\neq (a_1\cdot b_1+1)+(a_2\cdot b_2+1)=f(v_1)+f(v_2),$$ thus $f$ is not a linear transformation.
N.B. Although it isn't in your question, I want to take an opportunity to highlight another common mistake and point out that linear transformation is not the same as linear function. Indeed, a linear transformation is a function that necessarily passes through the origin, because of $T(0) = T(0 + 0) = T(0) + T(0) = 0$, while arbitrary linear function doesn't have this property.