<P> where Q (\ displaystyle Q) is the quantity of output and X 1, X 2, X 3,..., X n (\ displaystyle X_ (1), X_ (2), X_ (3), \ dotsc, X_ (n)) are the quantities of factor inputs (such as capital, labour, land or raw materials). </P> <P> If Q (\ displaystyle Q) is not a matrix (i.e., a scalar, a vector, or even a diagonal matrix), then this form does not encompass joint production, which is a production process that has multiple co-products . On the other hand, if f (\ displaystyle f) maps from R n k (\ displaystyle R ^ (n ^ (k))) then it is a joint production function expressing the determination of k (\ displaystyle k) different types of output based on the joint usage of the specified quantities of the n (\ displaystyle n) inputs . </P> <P> One formulation, unlikely to be relevant in practice, is as a linear function: </P> <Dl> <Dd> Q = a + b X 1 + c X 2 + d X 3 + ⋯ (\ displaystyle Q = a + bX_ (1) + cX_ (2) + dX_ (3) + \ dotsb) </Dd> </Dl>

In the illustration above which figure shows an aggregate production function