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This image presents a problem related to two different car rental pricing models, represented by two functions. Let's break down the tasks given:

### Problem Breakdown:
- **Tariff A:** It charges €0.75 per kilometer, with a fixed fee of €30.
- **Tariff B:** It's graphically represented by a line passing through two points: \( M_1(0, 0) \) and \( M_2(60, 90) \). 

We are tasked with the following:

### 1. Express \( f(x) \) and \( g(x) \) in terms of \( x \), where:
   - \( f(x) \) represents the cost under Tariff A.
   - \( g(x) \) represents the cost under Tariff B.
   - We are asked to define the nature of these functions.

### Solution for \( f(x) \) (Tariff A):
- Tariff A charges €0.75 per kilometer plus a fixed fee of €30. Therefore:
  \[
  f(x) = 0.75x + 30
  \]
  where \( x \) is the number of kilometers traveled.

### Solution for \( g(x) \) (Tariff B):
- From the graph, Tariff B is represented by a line passing through points \( M_1(0, 0) \) and \( M_2(60, 90) \). To find \( g(x) \), we need the equation of the line:
  1. The slope \( m \) is calculated as:
     \[
     m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{90 - 0}{60 - 0} = 1.5
     \]
  2. Using the point-slope form, the line equation is:
     \[
     g(x) = 1.5x
     \]

### Nature of the Functions:
- Both \( f(x) \) and \( g(x) \) are **linear functions** because they both represent straight lines. The rate of change (slope) is constant for each function.

### 2. Graph Representation:
You would need to reproduce the graph with the functions \( f(x) = 0.75x + 30 \) and \( g(x) = 1.5x \). The graph should show these functions as straight lines with different slopes.

### 3. Comparison and Threshold:
- The question asks at what point Tariff B becomes more expensive than €30 and when it becomes more economical than Tariff A.
  - For Tariff B to surpass €30:
    Solve \( g(x) = 30 \):
    \[
    1.5x = 30 \quad \Rightarrow \quad x = 20 \text{ kilometers}
    \]
    Thus, Tariff B surpasses €30 after 20 kilometers.

  - To find when Tariff A becomes more expensive than Tariff B:
    Solve \( f(x) = g(x) \):
    \[
    0.75x + 30 = 1.5x \quad \Rightarrow \quad 30 = 0.75x \quad \Rightarrow \quad x = 40 \text{ kilometers}
    \]
    So, Tariff B becomes cheaper after 40 kilometers.

Would you like more details or have any further questions on this? Here are five related questions to explore further:

1. How do you interpret the slopes of the two functions in terms of real-world meaning?
2. What would happen to the graph if the fixed fee in Tariff A changed to €40?
3. How can you visually determine where the two tariffs intersect on the graph?
4. What is the significance of the y-intercept for each tariff?
5. How does the steepness of the line for Tariff B affect cost over long distances?

**Tip:** When comparing costs over different distances, always look for the point of intersection between two pricing functions to determine which becomes more economical.

An enterprise offers two car rental pricing models for trips under 60 kilometers. Tariff A costs €0.75 per kilometer plus a mandatory fixed fee of €30, while Tariff B is represented graphically by a line passing through points M1(0, 0) and M2(60, 90). The task involves expressing f(x) and g(x), constructing graphs, and determining when Tariff A becomes more favorable than Tariff B.

Explore the comparison of two linear pricing models for car rentals: Tariff A with a fixed fee and per-kilometer rate, and Tariff B represented graphically. Solve for the break-even point and determine the favorable tariff.

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