The problem asks to estimate the area under the function $f(x)$ between $x = 0$ and $x = 4$, with the shaded region provided on the graph.

Step 1: Understanding the Graph

• The function appears to be a straight line, which likely represents a linear function.
• It rises from the origin (0,0) to the point (4,8), so we can infer that the equation of the line is $f(x) = 2x$.

Step 2: Calculating the Area

The shaded region forms a right triangle beneath the line from $x = 0$ to $x = 4$.

• The base of the triangle is 4 units (from $x = 0$ to $x = 4$).
• The height of the triangle is 8 units (the value of $f(x)$ at $x = 4$, which is $2 \times 4 = 8$).

The area $A$ of a triangle is given by the formula:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

Substituting the values:

$A = \frac{1}{2} \times 4 \times 8 = 16$

Step 3: Verifying the Answer Choices

The area is 16, but none of the choices directly match 16. However, choice (a) is $\frac{16}{3}$, which might appear as a close distractor, and (d) is 8. After reviewing the context, there might be confusion regarding interpretation, but based on direct calculation, none of the choices directly match the area.

Do you have any questions or want further clarification on how the choices are related?

Here are five related questions to deepen understanding:

1. How do you derive the equation of a line from a graph?
2. What is the relationship between a linear function and its geometric representation?
3. Why do we use the area formula for a triangle when calculating the area under a linear function?
4. How would the area change if the slope of the line increased?
5. Can this method be applied to nonlinear functions like $f(x) = x^2$?

Tip: Always double-check the function's equation from the graph to ensure accurate area estimation!