1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} -6 & \text{for } x < 2 \\ x - 7 & \text{for } x > 5 \end{cases}$$ We want to understand and analyze this function. 2. **Explain the function:** This function has two parts: - For values of $x$ less than 2, $f(x)$ is constant at $-6$. - For values of $x$ greater than 5, $f(x)$ is a linear function $x - 7$. 3. **Important notes:** The function is not defined for $2 \leq x \leq 5$ based on the given information. 4. **Evaluate the function at some points:** - For $x=1$ (which is less than 2), $f(1) = -6$. - For $x=6$ (which is greater than 5), $f(6) = 6 - 7 = -1$. 5. **Graph behavior:** - For $x < 2$, the graph is a horizontal line at $y = -6$. - For $x > 5$, the graph is a line with slope 1 and y-intercept $-7$. 6. **Summary:** The function is piecewise with a constant segment and a linear segment, with a gap between $x=2$ and $x=5$ where the function is not defined.