To solve this, let's break it down step-by-step using the information provided:

1. Domain of the function $f$:
   The domain of a function is the set of all possible $x$-values for which the function is defined. Since $f$ is a polynomial of degree 4, it is defined for all real numbers unless otherwise indicated by a graph with restrictions. Therefore, the domain of $f$ would typically be $(-\infty, \infty)$, unless the graph suggests limits at certain points.

2. Range of the function $f$:
   The range refers to the set of all possible $y$-values that the function can take. To determine the range from the graph, observe the highest and lowest points on the curve. If the graph of the 4th-degree polynomial has both a maximum and a minimum, the range will be between those points (inclusive).

   • If the graph rises indefinitely on both ends, the range is $[y_{\text{min}}, \infty)$.
   • If the graph falls indefinitely on both ends, the range is $(-\infty, y_{\text{max}}]$.

3. Checking the graph for the correct intervals:

   • For the domain, if the graph extends indefinitely in both the positive and negative $x$-directions, the domain will be $(-\infty, \infty)$.
   • For the range, examine the highest and lowest points on the graph. If the function reaches a minimum at some point, that minimum is the lower bound of the range. Similarly, the highest point on the graph (if there is one) will indicate the upper bound.

Once you analyze the graph carefully for these features, you can provide the domain and range using interval notation.

Follow-up questions:

1. What is the general shape of a degree-4 polynomial function?
2. How can the end behavior of the graph help determine the range?
3. What are the turning points in a polynomial, and how do they relate to the degree of the polynomial?
4. How does symmetry (if present) in the graph affect the domain and range?
5. If a function’s graph does not have endpoints, how do we describe its domain and range?

Tip: The domain of polynomial functions is typically all real numbers unless there are breaks or restrictions in the graph.