1. **State the problem:** We are given the function $f(x) = \sqrt{3 - x} - 1$ and need to understand its behavior and plot its values. 2. **Formula and domain:** The function involves a square root, so the expression inside must be non-negative: $$3 - x \geq 0$$ which simplifies to $$x \leq 3$$ This means the domain of $f$ is all real numbers $x$ such that $x \leq 3$. 3. **Evaluate the function at some points:** - At $x = 3$: $$f(3) = \sqrt{3 - 3} - 1 = \sqrt{0} - 1 = 0 - 1 = -1$$ - At $x = 2$: $$f(2) = \sqrt{3 - 2} - 1 = \sqrt{1} - 1 = 1 - 1 = 0$$ - At $x = 0$: $$f(0) = \sqrt{3 - 0} - 1 = \sqrt{3} - 1 \approx 1.732 - 1 = 0.732$$ - At $x = -1$: $$f(-1) = \sqrt{3 - (-1)} - 1 = \sqrt{4} - 1 = 2 - 1 = 1$$ 4. **Behavior:** The function decreases as $x$ increases towards 3, starting from values near 1 at $x = -1$ and going down to $-1$ at $x=3$. 5. **Summary:** The function $f(x) = \sqrt{3 - x} - 1$ is defined for $x \leq 3$, and its values range from $-1$ (at $x=3$) upwards.