Here is a detailed breakdown of the problem parts shown in the image:

Part (a): Graph the function

The function $f(x)$ is defined piecewise as:

• $1 - x$, if $x < -2$
• $x + 6$, if $-2 \leq x < 2$
• $-3$, if $x \geq 2$

To graph this function:

1. For $x < -2$: The function $f(x) = 1 - x$ is a linear equation with slope $-1$ and y-intercept $1$. Plot this segment to the left of $x = -2$ (not including $x = -2$).
2. For $-2 \leq x < 2$: The function $f(x) = x + 6$ is also linear, with slope $1$ and y-intercept $6$. This part is valid between $x = -2$ (included) and $x = 2$ (not included). Include a solid dot at $x = -2$ and an open circle at $x = 2$.
3. For $x \geq 2$: The function is constant $f(x) = -3$. Plot a horizontal line at $y = -3$, starting at $x = 2$ with a solid dot and extending right.

Part (b): Evaluate $\lim_{x \to -2} f(x)$

To determine the limit as $x \to -2$:

• From the left-hand side ($x \to -2^-$), the function follows $1 - x$, and substituting $x = -2$ gives $1 - (-2) = 3$.
• From the right-hand side ($x \to -2^+$), the function follows $x + 6$, and substituting $x = -2$ gives $(-2) + 6 = 4$.

Since the left-hand limit ($3$) and right-hand limit ($4$) are not equal, the limit does not exist.

Part (c): Evaluate $\lim_{x \to 1} f(x)$

To determine the limit as $x \to 1$:

• For $-2 \leq x < 2$, the function is $f(x) = x + 6$.
• Substituting $x = 1$ into $f(x) = x + 6$ gives $1 + 6 = 7$.

Thus, $\lim_{x \to 1} f(x) = 7$.

Part (d): Is $f$ continuous at $x = -2$?

A function is continuous at $x = -2$ if:

1. $\lim_{x \to -2} f(x)$ exists.
2. $f(-2)$ is defined.
3. $\lim_{x \to -2} f(x) = f(-2)$.

Here:

• $\lim_{x \to -2} f(x)$ does not exist (as shown in part (b)).
• $f(-2) = (-2) + 6 = 4$ (using the second piece).

Since the limit does not exist, $f(x)$ is not continuous at $x = -2$.

Let me know if you'd like a graph or have any specific questions! Here are five related questions to expand on this problem:

1. How do you determine the left-hand and right-hand limits for piecewise functions?
2. What are the conditions for a function to be continuous at a specific point?
3. How can the behavior of piecewise functions affect their limits?
4. What are some key differences between a limit existing and a function being continuous?
5. How would the function change if the second piece included $x = 2$?

Tip: When working with piecewise functions, always pay close attention to open and closed intervals and how they impact limits and continuity!