1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} 2x + 9 & \text{for } x < -3 \\ 2x - 2 & \text{for } x > 4 \end{cases}$$ 2. **Understand the function:** This function has two linear pieces defined on two separate intervals: one for $x < -3$ and one for $x > 4$. There is no definition for $-3 \leq x \leq 4$. 3. **Graphing each piece:** - For $x < -3$, the function is $f(x) = 2x + 9$. This is a line with slope 2 and y-intercept 9. - For $x > 4$, the function is $f(x) = 2x - 2$. This is a line with slope 2 and y-intercept -2. 4. **Evaluate key points:** - At $x = -3$, the first piece is not defined (strictly less than -3), but we can find the limit from the left: $$f(-3^-) = 2(-3) + 9 = -6 + 9 = 3$$ - At $x = 4$, the second piece is defined for $x > 4$, so not at 4 exactly, but the limit from the right is: $$f(4^+) = 2(4) - 2 = 8 - 2 = 6$$ 5. **Summary:** The graph consists of two separate line segments: - Line segment $y = 2x + 9$ for $x < -3$. - Line segment $y = 2x - 2$ for $x > 4$. There is a gap between $x = -3$ and $x = 4$ where the function is not defined. **Final answer:** The piecewise function is correctly defined as above with two linear pieces and a gap in between.