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

First write C in standard quadratic form: C(x) = 0.345x^2 − 7x + 200 (so a = 0.345, b = −7, c = 200).

Domain
- Mathematically, a quadratic is defined for all real x, so the domain would be (−∞, ∞).
- In the production context x is the number of units and cannot be negative, so the realistic domain is x ≥ 0 (and if you require whole units, x ∈ {0,1,2,...}).

Range
- The parabola opens upward (a = 0.345 > 0), so C has a minimum at its vertex.
- Vertex x-coordinate: xv = −b/(2a) = 7/(2·0.345) = 7/0.69 ≈ 10.1449.
- Minimum cost: Cmin = c − b^2/(4a) = 200 − 49/(4·0.345) ≈ 200 − 35.5073 ≈ 164.4927.

Thus for the realistic continuous domain x ≥ 0 the range is [≈164.4927, ∞). If x must be an integer, the minimum attainable cost occurs at x = 10 giving C(10) = 164.5, so the discrete range is {C(0), C(1), C(2), …} with minimum 164.5.