1. **State the problem:** We need to find the domain and range of the given function, determine if it is invertible, justify the invertibility, and if invertible, find the domain and range of its inverse. 2. **Analyze the graph:** The function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=2$. 3. **Domain:** The function is defined for all $x$ except where the vertical asymptote is, so the domain is $$(-\infty,0) \cup (0,\infty).$$ 4. **Range:** The function approaches $y=2$ but never touches it, and it goes up to near $y=10$ and down towards $2$ from above and below. The range is $$ (2, \infty).$$ 5. **Invertibility:** A function is invertible if it is one-to-one (passes the horizontal line test). Since the function is strictly decreasing on each side of the vertical asymptote and the two parts do not overlap in $y$-values, it is one-to-one. 6. **Justification sentence:** The function is invertible because it is one-to-one. 7. **Inverse domain and range:** The domain of the inverse is the range of the original function, and the range of the inverse is the domain of the original function. - Domain of inverse: $$ (2, \infty) $$ - Range of inverse: $$ (-\infty,0) \cup (0,\infty) $$ **Final answers:** - Domain of function: $$(-\infty,0) \cup (0,\infty)$$ - Range of function: $$(2, \infty)$$ - The function is invertible because it is one-to-one. - Inverse domain: $$(2, \infty)$$ - Inverse range: $$(-\infty,0) \cup (0,\infty)$$