Question 1160761: The function expresses the total cost (in dollars) of manufacturing units of product. If the maximum number of units which can be produced equals 50,000, state the restricted domain and range for this cost function.
Answer by CPhill(2303)   (Show Source):
You can put this solution on YOUR website! It appears that the specific algebraic formula for the cost function (e.g., $C(x)$) was omitted from your prompt. However, we can determine the restricted domain and define the exact range based on standard cost functions.
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### 1. Restricted Domain
Let $x$ represent the number of units of the product manufactured.
* Since production cannot be negative, the minimum number of units is $0$.
* The problem states the maximum capacity is $50{,}000$ units.
$$\mathbf{\text{Domain: }} [0,\, 50{,}000] \quad \text{or} \quad 0 \le x \le 50{,}000$$
*(If units must be whole numbers, the domain is the set of integers $x \in \{0, 1, 2, \dots, 50{,}000\}$).*
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### 2. Restricted Range
Let $C(x)$ denote the total cost function in dollars. The range consists of all possible total costs resulting from production levels within the restricted domain $[0, 50{,}000]$.
For standard cost functions where cost increases with production (monotonic non-decreasing):
* **Minimum Cost (lower bound):** Evaluated at $x = 0$ (representing the fixed overhead costs):
$$\text{Min Cost} = C(0)$$
* **Maximum Cost (upper bound):** Evaluated at full capacity $x = 50{,}000$:
$$\text{Max Cost} = C(50{,}000)$$
$$\mathbf{\text{Range: }} [C(0),\, C(50{,}000)] \quad \text{or} \quad C(0) \le C(x) \le C(50{,}000)$$
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### Example with Common Cost Functions
* **If $C(x) = mx + b$ (Linear Cost):**
* $C(0) = b$ (fixed cost)
* $C(50{,}000) = 50{,}000m + b$
* **Range:** $[b,\, 50{,}000m + b]$
*(For instance, if the function is $C(x) = 5x + 100{,}000$, the range is $[100{,}000,\, 350{,}000]$).*
To find the numerical range, plug $x = 0$ and $x = 50{,}000$ into the specific formula for $C(x)$.
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