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Let's solve the problem step by step.

### Given:
The total taxi charge $ C(x) $ for a ride that travels $ x $ miles is modeled by the function:
\[
C(x) = 3.5x + 3
\]
where:
- $3$ represents the initial fee.
- $3.5$ represents the additional cost per mile driven.

### Part (a): **Evaluate $ C(4) $**

To evaluate $ C(4) $, we substitute $ x = 4 $ into the function:

\[
C(4) = 3.5(4) + 3 = 14 + 3 = 17
\]

So, **$ C(4) = 17 $**.

### Part (b): **Interpret the result in the context**

The result $ C(4) = 17 $ means that a taxi ride for 4 miles will cost **$17**. This includes the initial fee of $3 and the cost per mile driven.

### Part (c): **Write a function equation for $ C^{-1} $, the inverse of $ C $**

To find the inverse function, we will reverse the operations in the function $ C(x) $.

1. Start with the equation $ y = 3.5x + 3 $.
2. Swap $ y $ and $ x $: 
   \[
   x = 3.5y + 3
   \]
3. Solve for $ y $:
   \[
   x - 3 = 3.5y
   \]
   \[
   y = \frac{x - 3}{3.5}
   \]

Thus, the inverse function is:
\[
C^{-1}(x) = \frac{x - 3}{3.5}
\]

### Part (d): **Evaluate $ C^{-1}(24) $**

To evaluate $ C^{-1}(24) $, substitute $ x = 24 $ into the inverse function:
\[
C^{-1}(24) = \frac{24 - 3}{3.5} = \frac{21}{3.5} = 6
\]

So, **$ C^{-1}(24) = 6 $**.

### Part (e): **Interpret the result in the context**

The result $ C^{-1}(24) = 6 $ means that if the total taxi fare is $24, the distance traveled was **6 miles**.

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Let me know if you'd like further clarifications on any part.

Here are 5 related questions to expand on this topic:
1. What would the cost of a 10-mile ride be according to the function $ C(x) $?
2. How would the function change if the initial fee increased to $5?
3. If a rider pays $50, how far did they travel according to the inverse function?
4. What does the slope of 3.5 represent in the function $ C(x) $?
5. How would you graph the function $ C(x) = 3.5x + 3 $?

**Tip:** Inverse functions help us reverse the relationship between input and output in real-life scenarios, such as finding the distance from the cost.

A New York City Taxi charges $3 initial fee plus $0.7 per 1/5 of a mile driven. The total taxi charge C(x) for a ride that travels x miles can be modeled by the function C(x) = 3.5x + 3. a) Evaluate C(4). b) Interpret the result in the context. c) Write a function equation for C^{-1}, the inverse of C. d) Evaluate C^{-1}(24). e) Interpret the result in the context.

Learn how to calculate the total fare for a taxi ride using the linear function C(x) = 3.5x + 3, and how to find the inverse function to determine the distance traveled based on the fare paid. Suitable for high school students, covering linear and inverse functions.

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