Part 1: Domain

The function is defined in two pieces:

• $f(x) = \frac{1}{6}x$ for $x < 0$
• $f(x) = x + 4$ for $x \geq 0$

This means the function is defined for all real numbers $x$, since there is no restriction on the values of $x$. Therefore, the domain is:

• All real numbers, which is $(-\infty, \infty)$.

The correct answer for domain is: B. $(-\infty, \infty)$

Part 2: Range

Let's analyze the output (range) of each piece:

1. For $x < 0$, $f(x) = \frac{1}{6}x$. As $x \to 0^-$ (approaching 0 from the left), $f(x) \to 0$. As $x \to -\infty$, $f(x) \to -\infty$. So, the range of this piece is $(-\infty, 0)$.
2. For $x \geq 0$, $f(x) = x + 4$. For $x = 0$, $f(x) = 4$, and as $x \to \infty$, $f(x) \to \infty$. Therefore, the range of this piece is $[4, \infty)$.

The combined range is the union of these two intervals:

• $(-\infty, 0) \cup [4, \infty)$.

The correct answer for range is: D. $(-\infty, 0] \cup [4, \infty)$

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Follow-up Questions:

1. How do you calculate the range of piecewise functions?
2. What happens to the range if the slopes or constants in each piece are modified?
3. How do you find the domain of a function if it's defined with multiple conditions?
4. Can a function's range be non-continuous like in this case? Why?
5. What is the significance of the boundary value $x = 0$ in this function?

Tip:

When determining the domain and range of a piecewise function, carefully examine each piece and their respective intervals to ensure proper inclusion or exclusion of values at boundaries.