1. **Problem:** Find the domain and range of the rational function $$f(x) = \frac{x+3}{x-4}$$. 2. **Domain:** The domain of a rational function is all real numbers except where the denominator is zero. 3. Set denominator equal to zero: $$x - 4 = 0 \implies x = 4$$. 4. So, the domain is all real numbers except $$x = 4$$, or in interval notation: $$(-\infty, 4) \cup (4, \infty)$$. 5. **Range:** To find the range, solve for $$x$$ in terms of $$y$$: $$y = \frac{x+3}{x-4}$$ Multiply both sides by $$x-4$$: $$y(x-4) = x + 3$$ Distribute $$y$$: $$yx - 4y = x + 3$$ Group $$x$$ terms on one side: $$yx - x = 4y + 3$$ Factor $$x$$: $$x(y - 1) = 4y + 3$$ Solve for $$x$$: $$x = \frac{4y + 3}{y - 1}$$ 6. The expression is undefined when the denominator $$y - 1 = 0$$, so $$y \neq 1$$. 7. Therefore, the range is all real numbers except $$y = 1$$, or $$(-\infty, 1) \cup (1, \infty)$$. **Final answer:** - Domain: $$(-\infty, 4) \cup (4, \infty)$$ - Range: $$(-\infty, 1) \cup (1, \infty)$$