Question

the daily production cost, c, for x units is modeled by the equation: c = 200 – 7x 0.345x2 explain how to find the domain and range of c.

Answer 1

This is a quadratic function, which is usually defined on all reals. But, the domain is restricted to  x≥0 because it does not make sense to produce a negative number of units.

Find the vertex of the parabola at

(10.14, 164.5).

Since the parabola opens up, the range is y ≥ 164.5

Answer 2

C(x) = 200 - 7x + 0.345x^2

Domain is the set of x-values (i.e. units produced) that are feasible. This is all the positive integer values + 0, in case that you only consider that can produce whole units.

Range is the set of possible results for c(x), i.e. possible costs.

You can derive this from the fact that c(x) is a parabole and you can draw it, for which you can find the vertex of the parabola, the roots, the y-intercept, the shape (it open upwards given that the cofficient of x^2 is positive). Also limit the costs to be positive.

You can substitute some values for x to help you, for example:

x      y
0    200
1    200 -7 +0.345 = 193.345
2    200 - 14 + .345 (4) = 187.38
3    200 - 21 + .345(9) = 182.105
4    200 - 28 + .345(16) = 177.52
5    200 - 35 + 0.345(25) = 173.625
6    200 - 42 + 0.345(36) = 170.42

10  200 - 70 + 0.345(100) =164.5
11 200 - 77 + 0.345(121) = 164.745
 
 
The functions does not have real roots, then the costs never decrease to 0.

The function starts at c(x) = 200, decreases until the vertex, (x =10, c=164.5) and starts to increase.

Then the range goes to 164.5 to infinity, limited to the solutcion for x = positive integers.