1. **State the problem:** Find the domain and range of the function $$f(x) = \frac{7x}{9x - 1}$$. 2. **Find the domain:** The domain consists of all real numbers $x$ for which the function is defined. The function is undefined when the denominator is zero. Set the denominator equal to zero and solve: $$9x - 1 = 0$$ $$9x = 1$$ $$x = \frac{1}{9}$$ So, the function is undefined at $x = \frac{1}{9}$. **Domain:** All real numbers except $x = \frac{1}{9}$. 3. **Find the range:** To find the range, solve for $x$ in terms of $y$ where $y = f(x)$: $$y = \frac{7x}{9x - 1}$$ Multiply both sides by the denominator: $$y(9x - 1) = 7x$$ Distribute $y$: $$9xy - y = 7x$$ Group terms with $x$ on one side: $$9xy - 7x = y$$ Factor out $x$: $$x(9y - 7) = y$$ Solve for $x$: $$x = \frac{y}{9y - 7}$$ The function is undefined when the denominator of this expression is zero: $$9y - 7 = 0$$ $$9y = 7$$ $$y = \frac{7}{9}$$ So, $y = \frac{7}{9}$ is not in the range. **Range:** All real numbers except $y = \frac{7}{9}$. 4. **Summary:** - Domain: $\{x \in \mathbb{R} \mid x \neq \frac{1}{9}\}$ - Range: $\{y \in \mathbb{R} \mid y \neq \frac{7}{9}\}$