1. **State the problem:** We want to analyze the function $$y = |x| + \frac{1}{|\sin(4x)|}$$ and understand its behavior. 2. **Recall important rules:** - The absolute value function $|x|$ outputs the non-negative value of $x$. - The sine function $\sin(4x)$ oscillates between $-1$ and $1$. - The denominator $|\sin(4x)|$ must not be zero because division by zero is undefined. 3. **Domain considerations:** - The function is undefined where $\sin(4x) = 0$. - Solve $\sin(4x) = 0$ for $x$: $$\sin(4x) = 0 \implies 4x = k\pi, \quad k \in \mathbb{Z} \implies x = \frac{k\pi}{4}$$ - So, the function is undefined at $x = \frac{k\pi}{4}$ for all integers $k$. 4. **Behavior near undefined points:** - As $x$ approaches $\frac{k\pi}{4}$, $|\sin(4x)|$ approaches zero, so $\frac{1}{|\sin(4x)|}$ tends to infinity. 5. **Summary:** - The function $y = |x| + \frac{1}{|\sin(4x)|}$ is defined for all real $x$ except at $x = \frac{k\pi}{4}$. - It has vertical asymptotes at these points. **Final answer:** The function is $$y = |x| + \frac{1}{|\sin(4x)|}$$ with domain $$\{x \in \mathbb{R} : x \neq \frac{k\pi}{4}, k \in \mathbb{Z}\}$$ and vertical asymptotes at these points.