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.