1. **Problem Statement:** Find the domain and range of the function $y = \frac{2x+3}{x-1}$. 2. **Domain:** The domain is all real numbers except where the denominator is zero because division by zero is undefined. 3. Set the denominator equal to zero and solve: $$x - 1 = 0$$ $$x = 1$$ 4. So, the domain is all real numbers except $x = 1$. 5. **Range:** To find the range, solve for $x$ in terms of $y$: $$y = \frac{2x+3}{x-1}$$ Multiply both sides by $x-1$: $$y(x-1) = 2x + 3$$ $$yx - y = 2x + 3$$ Bring all $x$ terms to one side: $$yx - 2x = y + 3$$ Factor out $x$: $$x(y - 2) = y + 3$$ Solve for $x$: $$x = \frac{y + 3}{y - 2}$$ 6. The expression for $x$ is undefined when the denominator is zero: $$y - 2 = 0$$ $$y = 2$$ 7. Therefore, the range is all real numbers except $y = 2$. **Final answers:** - Domain: $\{x \in \mathbb{R} \mid x \neq 1\}$ - Range: $\{y \in \mathbb{R} \mid y \neq 2\}$