Different types of cost functions (curves)

I see three types of cost functions:

1. $$C(q) = 4q$$ where $$q$$ is the quantity
2. $$C(w_1, w_2 ,q) = w_1w_2q$$ where $$w_1, w_2$$ are the rental rates and $$q$$ is the quantity of the product
3. $$C(\mathbf{w}, \mathbf{x}) = \mathbf{w} \cdot \mathbf{x}$$ where $$\mathbf{w}$$ is the price vector and $$\mathbf{x}$$ denotes the input factors

Can anyone tell me what each of these mean? The second one is probably called the conditional cost function and the third one comes from the definition of cost. What's the first one? I have seen it being used in perfect competition and monopoly for instance, but what's the difference between 1 and 2 then? We can also use (2) in perfect competition and monopoly I think, or can we not? I am confused.

Edit: I think (1) and (2) are the same. They both are probably conditional cost functions. (2) probably considers the prices as values (with the assumption that prices aren't changing). (3) is the generic definition or the unconditional cost function. When optimized to produce quantity $$q$$, (3) produces (1) and (2).

1. $$C(q)$$: given fixed input prices, this is the cost of producing $$q$$ units, assuming the use of inputs is optimal given the technology and the fixed input prices.
2. $$C(w_1, w_2 ,q)$$: same as above, but the function is more general because the cost is given for all $$w_1, w_2$$ input prices, not just some specific fixed ones.
3. $$C(\mathbf{w}, \mathbf{x})$$: I have never seen this notation. $$C(\mathbf{w},q)$$ is the solution to the minimalization problem $$\min_{\mathbf{x}} \mathbf{w} \cdot \mathbf{x}$$ $$\text{s.t.:} \ \ f(\mathbf{x}) = q.$$
• What do we use for something like $C = wr + kl$? I was denoting that with the third notation. Oct 13, 2022 at 14:39
• What do mean "What do we use for"? Your purpose is not clear to me. $\mathbf{w} \cdot \mathbf{x}$ is good for some uses, $C(\mathbf{x},q)$ is good for others. Oct 13, 2022 at 14:39
• Like, what notation do we use for $C = wr + kl$? Don't we use $C(w, r, k, l)$? I was denoting this as $C(\mathbf{w},\mathbf{x})$ where $\mathbf{w}$ is the price vector and $\mathbf{x}$ is the input vector. Oct 13, 2022 at 14:40
• What notation do we use for $x_1\cdot \alpha^y - \gamma$...? I don't see why every formula would have its function. Is there something wrong with writing $wr+kl$? Oct 13, 2022 at 14:43