Range Check 0333F8

1. **State the problem:** We need to determine if each piecewise function $f(x)$, $h(x)$, and $g(x)$ has a range of $(-\infty,4)$. This means the output values of the function cover all real numbers less than 4 but do not include 4 or any number greater than 4.\n\n2. **Recall the range of piecewise linear functions:** The range is the set of all possible output values. For linear pieces, the range depends on the behavior of each piece and the domain restrictions.\n\n---\n\n### Part a: $f(x) = \begin{cases} x - 3, & x < -2 \\ 5 - x, & x > 1 \end{cases}$\n\n3. **Analyze first piece:** $y = x - 3$ for $x < -2$. As $x \to -\infty$, $y \to -\infty$. At $x = -2$, $y = -2 - 3 = -5$. So this piece covers $(-\infty, -5)$.\n\n4. **Analyze second piece:** $y = 5 - x$ for $x > 1$. As $x \to 1^+$, $y \to 5 - 1 = 4$. As $x \to \infty$, $y \to -\infty$. So this piece covers $(-\infty, 4)$.\n\n5. **Combine ranges:** The first piece covers $(-\infty, -5)$ and the second covers $(-\infty, 4)$. The union is $(-\infty, 4)$.\n\n6. **Conclusion for a:** Yes, $f(x)$ has range $(-\infty,4)$.\n\n---\n\n### Part b: $h(x) = \begin{cases} x - 1, & x < -1 \\ 3 - x, & x > 2 \end{cases}$\n\n7. **First piece:** $y = x - 1$ for $x < -1$. As $x \to -\infty$, $y \to -\infty$. At $x = -1$, $y = -1 - 1 = -2$. So range is $(-\infty, -2)$.\n\n8. **Second piece:** $y = 3 - x$ for $x > 2$. As $x \to 2^+$, $y \to 3 - 2 = 1$. As $x \to \infty$, $y \to -\infty$. So range is $(-\infty, 1)$.\n\n9. **Combine ranges:** Union is $(-\infty, 1)$. This does not cover values between 1 and 4.\n\n10. **Conclusion for b:** No, $h(x)$ does not have range $(-\infty,4)$.\n\n---\n\n### Part c: $g(x) = \begin{cases} x + 5, & x < -1 \\ -x - 5, & x > 1 \end{cases}$\n\n11. **First piece:** $y = x + 5$ for $x < -1$. As $x \to -\infty$, $y \to -\infty$. At $x = -1$, $y = -1 + 5 = 4$. So range is $(-\infty, 4)$.\n\n12. **Second piece:** $y = -x - 5$ for $x > 1$. As $x \to 1^+$, $y \to -1 - 5 = -6$. As $x \to \infty$, $y \to -\infty$. So range is $(-\infty, -6)$.\n\n13. **Combine ranges:** Union is $(-\infty, 4)$ because the first piece already covers up to 4.\n\n14. **Conclusion for c:** Yes, $g(x)$ has range $(-\infty,4)$.