1. **State the problem:** Find the domain, range, and inverse of the function $f(x) = \frac{2x + 4}{1 - 2x}$. 2. **Find the domain:** The domain consists of all $x$ values for which the function is defined. The denominator must not be zero. Set the denominator not equal to zero: $$1 - 2x \neq 0$$ Solve for $x$: $$2x \neq 1 \implies x \neq \frac{1}{2}$$ So, the domain is all real numbers except $x = \frac{1}{2}$. 3. **Find the range:** Let $y = f(x) = \frac{2x + 4}{1 - 2x}$. To find the range, solve for $x$ in terms of $y$: $$y = \frac{2x + 4}{1 - 2x}$$ Multiply both sides by $1 - 2x$: $$y(1 - 2x) = 2x + 4$$ Distribute $y$: $$y - 2xy = 2x + 4$$ Group $x$ terms on one side: $$-2xy - 2x = 4 - y$$ Factor $x$: $$x(-2y - 2) = 4 - y$$ Solve for $x$: $$x = \frac{4 - y}{-2y - 2} = \frac{4 - y}{-(2y + 2)} = -\frac{4 - y}{2y + 2}$$ The inverse function exists except when the denominator is zero: $$2y + 2 \neq 0 \implies y \neq -1$$ Thus, the range is all real numbers except $y = -1$. 4. **Find the inverse:** Swap $x$ and $y$ in the previous expression for $x$: $$y = -\frac{4 - x}{2x + 2}$$ Alternatively, express the inverse function explicitly: From step 3, the inverse function is $$f^{-1}(x) = -\frac{4 - x}{2x + 2}$$ **Final answers:** - Domain: $\{x \in \mathbb{R} : x \neq \frac{1}{2}\}$ - Range: $\{y \in \mathbb{R} : y \neq -1\}$ - Inverse function: $$f^{-1}(x) = -\frac{4 - x}{2x + 2}$$