1. **State the problem:** Find the domain and range of the piecewise function $$g(x) = \begin{cases} 6x + 7, & x \leq -2 \\ 4 - 3x, & x > -2 \end{cases}$$ 2. **Domain:** The domain is all values of $x$ for which the function is defined. Here, the function is defined for all real numbers because it covers $x \leq -2$ and $x > -2$. **Domain:** $$(-\infty, \infty)$$ 3. **Range for $x \leq -2$:** Use the expression $6x + 7$. Calculate the value at $x = -2$: $$6(-2) + 7 = -12 + 7 = -5$$ As $x$ decreases (goes to $-\infty$), $6x + 7$ goes to $-\infty$. So, for $x \leq -2$, the range is: $$(-\infty, -5]$$ 4. **Range for $x > -2$:** Use the expression $4 - 3x$. Calculate the value as $x$ approaches $-2$ from the right: $$4 - 3(-2) = 4 + 6 = 10$$ As $x$ increases to $\infty$, $4 - 3x$ goes to $-\infty$. So, for $x > -2$, the range is: $$(-\infty, 10)$$ 5. **Combine the ranges:** The first piece covers $(-\infty, -5]$ and the second piece covers $(-\infty, 10)$. The overall range is the union: $$(-\infty, 10)$$ 6. **Final answer:** **Domain:** $$(-\infty, \infty)$$ **Range:** $$(-\infty, 10)$$