To determine the range of the rational function [tex]\( C(f) \)[/tex] representing the average monthly cost of owning a cell phone, let's break down the costs and analyze the function step-by-step.
1. Initial Costs and Monthly Charges:
- The cell phone has an initial cost of [tex]$300.
- Thereafter, the cost each month is $[/tex]40.
2. Total Cost Over Time:
- If we denote the number of months of ownership by [tex]\( f \)[/tex], the total cost after [tex]\( f \)[/tex] months is given by:
[tex]\[
\text{Total Cost} = 300 + 40f
\][/tex]
3. Average Monthly Cost:
- To find the average monthly cost, we divide the total cost by the number of months [tex]\( f \)[/tex]:
[tex]\[
C(f) = \frac{300 + 40f}{f}
\][/tex]
4. Simplification:
- Simplifying the function [tex]\( C(f) \)[/tex], we get:
[tex]\[
C(f) = \frac{300}{f} + 40
\][/tex]
5. Behavior Analysis:
- As [tex]\( f \)[/tex] increases, the term [tex]\(\frac{300}{f}\)[/tex] will become smaller and smaller.
- When [tex]\( f \)[/tex] is very large (approaches infinity), [tex]\(\frac{300}{f}\)[/tex] approaches 0.
- Hence, as [tex]\( f \to \infty \)[/tex], [tex]\( C(f) \)[/tex] approaches 40.
6. Minimum Value:
- The minimum value of [tex]\( C(f) \)[/tex] is the smallest it can get, which happens to be 40 as [tex]\( f \)[/tex] becomes very large.
7. Range Determination:
- As [tex]\( f \)[/tex] decreases from infinity, [tex]\( \frac{300}{f} \)[/tex] increases, contributing to the average cost.
- However, since [tex]\( \frac{300}{f} \)[/tex] can become arbitrarily large for very small values of [tex]\( f \)[/tex], there is no upper limit to the cost.
Therefore, the range of the function [tex]\( C(f) \)[/tex] is:
[tex]\[
[40, \infty)
\][/tex]
This falls neither into a negative range nor a limited positive range but rather reflects that the cost can start from 40 USD from an average perspective and increase indefinitely.
Hence, the correct answer is:
[tex]\[
R: [40, \infty)
\][/tex]
