Sketch the graph of the function. \( f(x)=\frac{3-x}{x-3} \)

To sketch the graph of the function \( f(x) = \frac{3 - x}{x - 3} \), we can start by simplifying the expression and analyzing its key features. 1. **Simplification**: The function can be rewritten as: \[ f(x) = \frac{-(x - 3)}{x - 3} = -1 \quad \text{for } x \neq 3 \] This indicates that the function is equal to \(-1\) for all \(x\) except at \(x = 3\). 2. **Domain**: The function is undefined at \(x = 3\) because it would result in division by zero. Therefore, the domain of \(f(x)\) is all real numbers except \(x = 3\): \[ \text{Domain: } x \in \mathbb{R}, x \neq 3 \] 3. **Horizontal Asymptote**: Since the function is constant at \(-1\) for all \(x\) except \(x = 3\), the horizontal line \(y = -1\) can be considered a horizontal asymptote. 4. **Vertical Asymptote**: There is a vertical asymptote at \(x = 3\) because the function approaches \(-1\) from both sides as \(x\) approaches \(3\), but it is undefined at that point. 5. **Graph**: - The graph will be a horizontal line at \(y = -1\) for all \(x\) except at \(x = 3\). - At \(x = 3\), there will be a hole in the graph (indicating that the function is undefined). ### Sketch of the Graph: - Draw a horizontal line at \(y = -1\). - Mark a hole (or an open circle) at the point \((3, -1)\) to indicate that the function is not defined at that point. Here is a rough sketch of the graph: ``` y | | ------------------- (y = -1) | o | / | / | / | / | / | / | / | / |/_________________________ x ``` In this sketch: - The horizontal line represents \(y = -1\). - The open circle at \((3, -1)\) indicates the hole in the graph at \(x = 3\). This completes the analysis and sketch of the function \(f(x) = \frac{3 - x}{x - 3}\).