1. **State the problem:** We have a piecewise function defined as: $$f(x) = \begin{cases} -3x + 4 & \text{for } x < 1 \\ -2x + 3 & \text{for } 1 \leq x \leq 5 \\ 3x - 24 & \text{for } x > 5 \end{cases}$$ We want to understand the behavior of this function, including its values at boundary points and continuity. 2. **Evaluate the function at boundary points:** - At $x=1$, since $1 \leq x \leq 5$, use $f(x) = -2x + 3$: $$f(1) = -2(1) + 3 = -2 + 3 = 1$$ - At $x=5$, since $1 \leq x \leq 5$, use $f(x) = -2x + 3$: $$f(5) = -2(5) + 3 = -10 + 3 = -7$$ 3. **Check left-hand limit at $x=1$:** For $x < 1$, $f(x) = -3x + 4$. Calculate the limit as $x \to 1^-$: $$\lim_{x \to 1^-} f(x) = -3(1) + 4 = -3 + 4 = 1$$ 4. **Check right-hand limit at $x=1$:** For $x \geq 1$, $f(x) = -2x + 3$. Calculate the limit as $x \to 1^+$: $$\lim_{x \to 1^+} f(x) = -2(1) + 3 = 1$$ Since left and right limits and the function value at $x=1$ are all equal to 1, the function is continuous at $x=1$. 5. **Check left-hand limit at $x=5$:** For $x \leq 5$, $f(x) = -2x + 3$. Calculate the limit as $x \to 5^-$: $$\lim_{x \to 5^-} f(x) = -2(5) + 3 = -7$$ 6. **Check right-hand limit at $x=5$:** For $x > 5$, $f(x) = 3x - 24$. Calculate the limit as $x \to 5^+$: $$\lim_{x \to 5^+} f(x) = 3(5) - 24 = 15 - 24 = -9$$ 7. **Check function value at $x=5$:** Since $x=5$ is included in the middle piece, $f(5) = -7$. 8. **Conclusion on continuity at $x=5$:** Left limit is $-7$, right limit is $-9$, and function value is $-7$. Since left and right limits are not equal, the function is discontinuous at $x=5$. 9. **Summary:** - The function is continuous at $x=1$. - The function is discontinuous at $x=5$. This analysis helps understand the piecewise function's behavior and continuity.