1. **State the problem:** We want to analyze the function $$y=|x|+\frac{1}{|\sin(4x)|}$$ which involves absolute values and a trigonometric function. 2. **Understand the components:** The function has two parts: the absolute value of $x$, which is always non-negative, and the reciprocal of the absolute value of $\sin(4x)$. 3. **Important rules:** - $|x|$ is the distance of $x$ from zero, so $|x| \geq 0$ for all $x$. - $|\sin(4x)|$ oscillates between 0 and 1. - The term $\frac{1}{|\sin(4x)|}$ is undefined where $\sin(4x) = 0$, i.e., at points $x = \frac{k\pi}{4}$ for integers $k$. 4. **Behavior near singularities:** At points where $\sin(4x) = 0$, the function tends to infinity because of division by zero. 5. **Summary:** The function is defined for all $x$ except $x = \frac{k\pi}{4}$, where it has vertical asymptotes. 6. **Graph features:** - The function is always positive. - It has vertical asymptotes at $x = \frac{k\pi}{4}$. - The function grows linearly with $|x|$ away from these points. **Final answer:** The function $$y=|x|+\frac{1}{|\sin(4x)|}$$ is defined for all $x \neq \frac{k\pi}{4}$, $k \in \mathbb{Z}$, with vertical asymptotes at these points and is always positive.