To find the domain and range of the function $f(x) = \frac{1}{4}(x^3 + 6x^2 + 5x - 4)$ shown in the graph, let's analyze both:

1. Domain:

The domain of a polynomial function, like this cubic function, is all real numbers since there are no restrictions such as division by zero or square roots of negative numbers. Therefore, the domain is: $\text{Domain: } (-\infty, \infty)$

2. Range:

The range of a cubic function can vary, and it typically covers all real numbers since it can increase or decrease without bound. From the graph, we see that the function moves upwards to positive infinity and downwards to negative infinity, indicating that the function covers all possible $y$-values. Thus, the range is: $\text{Range: } (-\infty, \infty)$

However, the graph provided shows specific intervals for the domain and range. The options in the image provide the domain as $(-6.5, 2.5)$. This suggests that the given range for this particular question might be focusing on a specific portion of the graph, rather than the entire function's domain and range.

Given this context and the information visible in the graph:

• If you are looking at the whole graph of the cubic function:
  • Domain: $(-\infty, \infty)$
  • Range: $(-\infty, \infty)$
• If the question refers to the portion shown:
  • Domain: $(-6.5, 2.5)$
  • Range: Check the vertical extent of the graph shown within this interval.

Would you like further details or have any other questions?

Follow-up Questions:

1. Would you like to know how to find the exact turning points of this cubic function?
2. Do you need a step-by-step explanation of how to determine the domain and range of other polynomial functions?
3. Would you like to explore how the coefficients of the polynomial affect the shape of the graph?
4. Do you need assistance with other types of functions, like rational or trigonometric functions?
5. Would you like to discuss how to approximate the range from the graph?

Tip:

For polynomial functions, the domain is always all real numbers, but the range can be determined by analyzing the function's behavior and its graph.