A firm's total cost function is given as:

TC = q³( 10 + 5q²)

Where:

TC = total cost
q = quantity produced

To find the average cost function, we need to first find the total cost for a given quantity produced. The average cost (AC) is calculated by dividing the total cost by the quantity produced.

Average Cost (AC) = TC / q

Substitute the given total cost function TC = q³(10 + 5q²) into the formula:

AC = (q³(10 + 5q²)) / q
AC = 10q + 5q²

Therefore, the firm's average cost function is AC = 10q + 5q².