https://file.aiquickdraw.com/mathbot/file/1726405389966kqcnnfw1.jpg

Let's analyze the graph and answer the questions step-by-step.

1. **Question 3.4.1**: Find \( x \) when \( f(x) = 4 \).
   - The function \( f(x) \) is the parabola, and we need to determine the \( x \)-values where \( f(x) = 4 \). From the graph, the parabola crosses \( y = 4 \) at two points. These points are approximately \( x = -1 \) and \( x = 3 \).

2. **Question 3.4.2**: Find the y-intercept of \( g \).
   - The function \( g(x) \) is the straight line. The y-intercept occurs where \( x = 0 \). Observing the graph, the line crosses the y-axis at \( y = 7 \).

3. **Question 3.4.3**: Find the x-value where \( f(x) = g(x) \) for \( x < 0 \).
   - We need to find where the two graphs intersect for \( x < 0 \). From the graph, the intersection point occurs around \( x = -2 \).

4. **Question 3.4.4**: Compute \( g(2) - f(2) \).
   - First, find \( g(2) \) from the graph. At \( x = 2 \), the line \( g(x) \) is approximately at \( y = 5 \).
   - Now, find \( f(2) \) from the graph. At \( x = 2 \), the parabola \( f(x) \) is at \( y = 0 \).
   - So, \( g(2) - f(2) = 5 - 0 = 5 \).

### Answers Summary:
- **3.4.1**: \( x = -1 \) and \( x = 3 \).
- **3.4.2**: The y-intercept of \( g \) is 7.
- **3.4.3**: The x-value where \( f(x) = g(x) \) for \( x < 0 \) is approximately \( x = -2 \).
- **3.4.4**: \( g(2) - f(2) = 5 \).

Would you like further clarification on any part of the solution? Here are some related questions for deeper understanding:

1. How can we determine the vertex of the parabola \( f(x) \)?
2. What is the equation of the line \( g(x) \)?
3. How can we algebraically solve for the points where \( f(x) = g(x) \)?
4. How do we compute the x-intercepts of the parabola \( f(x) \)?
5. What transformations can be applied to the function \( f(x) \) to change its graph?

**Tip**: When analyzing graphs, always look for key features like intercepts, turning points, and intersections to guide your solutions.

Use the graph to read off: 3.4.1) x if f(x) = 4, 3.4.2) The y-intercept of g, 3.4.3) The x-value where f(x) = g(x) for x < 0, 3.4.4) g(2) - f(2)

Learn how to analyze the graph of a quadratic function f(x) and a linear function g(x). This includes identifying key points such as intercepts, function values, and solving for x when f(x) = g(x). Suitable for Grades 10-12.

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