Sletontue graph of eain of the followins function [ d k(x)=(x+6)/(x^2)+4 ]

To sketch the graph of the function \( k(x) = \frac{x+6}{x^2+4} \), we need to analyze its key features:1. **Domain**: The function is defined for all real numbers because the denominator is never zero (since for all ).2. **Intercepts**: - **y-intercept**: Set : - **x-intercept**: Set \( k(x) = 0 \): 3. **Asymptotes**: - **Vertical asymptotes**: There are no vertical asymptotes because the denominator is never zero. - **Horizontal asymptote**: As , the degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is .4. **Behavior near intercepts and asymptotes**: - Near , the function crosses the x-axis. - As , \( k(x) \to 0 \).5. **First derivative** (for increasing/decreasing behavior): Analyze the sign of \( k'(x) \) to determine intervals of increase and decrease.6. **Second derivative** (for concavity and inflection points): Analyze the sign of \( k''(x) \) to determine concavity.By combining these analyses, you can sketch the graph of \( k(x) \). The graph will have a y-intercept at \( \left(0, \frac{3}{2}\right) \), an x-intercept at \( (-6, 0) \), and approach the horizontal asymptote as . The function will be smooth and continuous for all .