To determine the domain and range of the function $f(x) = \frac{1}{3}x - 4$, let's break it down:

Domain:

The domain refers to the set of all possible input values ($x$) for which the function is defined. Since $f(x) = \frac{1}{3}x - 4$ is a linear function, there are no restrictions on $x$ (e.g., division by zero, square roots of negative numbers, etc.), so the domain is all real numbers.

Thus, the domain is: $\boxed{(-\infty, \infty)}$

Range:

The range refers to the set of all possible output values ($y$ or $f(x)$) for the function. Since this is a linear function, as $x$ takes any real value, $f(x)$ will also take any real value (it has no upper or lower bound).

Thus, the range is: $\boxed{(-\infty, \infty)}$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Would you like more details or explanations about how linear functions behave? Let me know if you need further clarification!

Related questions:

1. What is the slope and y-intercept of the function $f(x) = \frac{1}{3}x - 4$?
2. How do you find the domain and range of a quadratic function?
3. What is the graph of the linear function $f(x) = \frac{1}{3}x - 4$ like?
4. How would the domain and range change if the function involved a square root or a rational expression?
5. What happens to the range of the function if we transform it (shift it up/down or scale it)?

Tip: For linear functions like this one, the domain and range are always $(-\infty, \infty)$, unless there is some constraint on $x$ or $y$.