1. **State the problem:** We want to understand the function $$y = 1 + \frac{1}{2 - |x|} - \frac{1}{2}$$ and how it behaves. 2. **Recall the absolute value:** The absolute value $|x|$ means the distance of $x$ from zero, so it is always non-negative. 3. **Rewrite the function:** Combine the constants first: $$y = 1 - \frac{1}{2} + \frac{1}{2 - |x|} = \frac{1}{2} + \frac{1}{2 - |x|}$$ 4. **Domain considerations:** The denominator $2 - |x|$ cannot be zero because division by zero is undefined. So, set: $$2 - |x| \neq 0 \implies |x| \neq 2$$ This means $x \neq 2$ and $x \neq -2$. 5. **Summary:** The function is $$y = \frac{1}{2} + \frac{1}{2 - |x|}$$ with domain all real numbers except $x = \pm 2$. 6. **Behavior near $x=\pm 2$:** As $x$ approaches $2$ or $-2$, the denominator $2 - |x|$ approaches zero, so $y$ tends to infinity or negative infinity depending on the direction. 7. **Graph features:** The function has vertical asymptotes at $x=2$ and $x=-2$. This explains the function and its key properties.