We are tasked with finding the domain and range of the function $f(x) = \sqrt[3]{x - 3}$.

Step 1: Analyze the domain

The cube root function $\sqrt[3]{x}$ is defined for all real numbers. For $\sqrt[3]{x - 3}$, the expression inside the cube root, $x - 3$, can also take any real value. Hence, there are no restrictions on $x$. The domain of $f(x)$ is: $\text{Domain: } (-\infty, \infty)$

Step 2: Analyze the range

The cube root function $\sqrt[3]{x}$ outputs all real numbers, as it is a continuous and one-to-one function. Similarly, $\sqrt[3]{x - 3}$ also outputs all real numbers since subtracting 3 from $x$ shifts the input but does not affect the overall behavior. The range of $f(x)$ is: $\text{Range: } (-\infty, \infty)$

Correct Answer:

The domain is $(-\infty, \infty)$ and the range is $(-\infty, \infty)$. This corresponds to the last option: $\text{Domain: } (-\infty, \infty), \text{ Range: } (-\infty, \infty).$

Would you like further explanation on domain and range or cube root functions? Here are some related questions to deepen understanding:

1. What is the difference between cube root and square root in terms of domain and range?
2. How does a horizontal shift (e.g., $x - 3$) affect the graph of a function?
3. Can you provide an example of a function with a restricted domain or range?
4. How do you determine the domain and range for composite functions?
5. Why do cube root functions have unrestricted domains and ranges, while square roots do not?

Tip: Always visualize the graph of a function when analyzing its domain and range—it can provide immediate insights!