Suppose that the cost for manufacturing
n
desks is
C
dollars, so we have
C
=
f
(
n
)
. Which of the following statements is the best interpretation of
f
(
2
n
)
?

The cost, in dollars, has doubled for manufacturing
n
desks.
The number of desks has doubled.
The cost, in dollars, of manufacturing twice as many desks.
The cost, in dollars, for manufacturing
n
desks has been cut in half.
The cost, in dollars, for manufacturing half as many desks.

Let's analyze the function \( C = f(n) \), where \( C \) is the cost in dollars for manufacturing \( n \) desks.

**Interpretation of \( f(2n) \):**

1. \( f(2n) \) means the function \( f \) is evaluated at \( 2n \). In other words, \( 2n \) represents twice the original number of desks. Thus, \( f(2n) \) gives the cost of manufacturing \( 2n \) desks, not just \( n \) desks.

Now, let's review each statement to find the best interpretation of \( f(2n) \):

1. **"The cost, in dollars, has doubled for manufacturing \( n \) desks."**
   - This is incorrect because \( f(2n) \) refers to the cost of manufacturing twice the number of desks, not just \( n \).

2. **"The number of desks has doubled."**
   - While this statement is true, it does not correctly interpret what \( f(2n) \) represents in terms of cost.

3. **"The cost, in dollars, of manufacturing twice as many desks."**
   - This statement is correct. \( f(2n) \) represents the cost of manufacturing twice the original number of desks, which is \( 2n \).

4. **"The cost, in dollars, for manufacturing \( n \) desks has been cut in half."**
   - This is incorrect because \( f(2n) \) does not describe a reduction in cost; it describes the cost for more desks.

5. **"The cost, in dollars, for manufacturing half as many desks."**
   - This is incorrect because \( f(2n) \) involves twice as many desks, not half.

**Best interpretation:**
- **"The cost, in dollars, of manufacturing twice as many desks."**

Would you like more details or have any other questions?

Here are some related questions to consider:

1. What would \( f(n/2) \) represent in this context?
2. If the cost function is linear, what would \( f(2n) \) be in terms of \( f(n) \)?
3. How would the interpretation change if the function \( f(n) \) were exponential?
4. What assumptions can we make about the function \( f(n) \) given that it's a cost function?
5. How does the behavior of \( f(n) \) change if there are fixed costs involved in manufacturing?

**Tip:** When interpreting functions, always match the variable input (e.g., \( n \), \( 2n \), etc.) to the real-world scenario it represents.

Understand the interpretation of functions in cost analysis with a focus on f(2n). Learn how f(2n) represents the cost of manufacturing twice as many desks compared to f(n). Explore detailed explanations and examples. Suitable for advanced high school students and beyond.

Math Problem Statement

Suppose that the cost for manufacturing n desks is C dollars, so we have C

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