If we produce 350 units of output, the total cost is $11,500. It costs us $12,900 to produce 420 units. Assuming this is linear, determine the cost function.

If by cost function you mean the unit cost or the variable unit cost then please elaborate otherwise divide each value by the quantity

The increase in cost, divided by the increase in output units, gives the unit variable cost.

Then knowing the uvc you can go back to either of the two observations to determine the fixed cost.

Finally, put it into a linear function of the form

C(q) = qx + F

where x is the uvc and F denotes fixed cost.

Back to you, mf158052 to wrap it up.