1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} -\frac{2}{3}x + 3, & x \leq -1 \\ 5, & x > -1 \end{cases}$$ We want to understand and solve for values or analyze this function. 2. **Analyze each piece:** - For $x \leq -1$, the function is linear: $f(x) = -\frac{2}{3}x + 3$. - For $x > -1$, the function is constant: $f(x) = 5$. 3. **Evaluate the function at the boundary $x = -1$:** $$f(-1) = -\frac{2}{3}(-1) + 3 = \frac{2}{3} + 3 = \frac{2}{3} + \frac{9}{3} = \frac{11}{3} \approx 3.67$$ 4. **Check continuity at $x = -1$:** - Left limit: $\lim_{x \to -1^-} f(x) = f(-1) = \frac{11}{3}$ - Right limit: $\lim_{x \to -1^+} f(x) = 5$ Since $\frac{11}{3} \neq 5$, the function has a jump discontinuity at $x = -1$. 5. **Summary:** - For $x \leq -1$, the graph is a line segment from $x = -7$ (or less) up to $x = -1$ with $f(x) = -\frac{2}{3}x + 3$. - At $x = -1$, the function value is $\frac{11}{3}$. - For $x > -1$, the function is constant at $5$. This matches the description of the graph with two parts: a line segment on the left and a horizontal line on the right. **Final answer:** The function is piecewise defined as above with a jump discontinuity at $x = -1$.