1. **State the problem:** We have a piecewise function $$g(x) = \begin{cases} x - 2 & x \leq 3 \\ x + 1 & 3 \leq x \leq 5 \end{cases}$$ We want to find the range $R_f$ of this function for $x < 5$. 2. **Understand the domain and function pieces:** - For $x \leq 3$, $g(x) = x - 2$. - For $3 \leq x \leq 5$, $g(x) = x + 1$. - The domain considered is $x < 5$ (so up to but not including 5). 3. **Find the range for each piece:** - For $x \leq 3$, the function is $g(x) = x - 2$. The maximum $x$ here is 3, so the maximum value is $3 - 2 = 1$. As $x$ goes to $-\infty$, $g(x)$ goes to $-\infty$. So the range for this piece is $(-\infty, 1]$. - For $3 \leq x < 5$, the function is $g(x) = x + 1$. At $x=3$, $g(3) = 4$. At $x$ approaching 5 from the left, $g(x)$ approaches $6$ but does not include 6 since $x<5$. So the range for this piece is $[4, 6)$. 4. **Combine the ranges:** The overall range is the union of $(-\infty, 1]$ and $[4, 6)$. 5. **Check for continuity or gaps:** There is a gap between 1 and 4, so the range is not continuous. **Final answer:** $$R_f = (-\infty, 1] \cup [4, 6)$$