c. \( \frac{-11}{x}, \frac{7}{x-4}, \frac{x}{x^{2}-16} \)

It looks like you've provided three rational functions: c. \( \frac{-11}{x} \), \( \frac{7}{x-4} \), \( \frac{x}{x^{2}-16} \) To analyze these functions, let's explore their **domains**, **vertical asymptotes**, and **horizontal asymptotes**. --- ### 1. \( \frac{-11}{x} \) **Domain:** - The denominator \( x \) cannot be zero. - **Domain:** \( x \neq 0 \) **Vertical Asymptote:** - Occurs where the denominator is zero. - \( x = 0 \) **Horizontal Asymptote:** - Compare the degrees of the numerator and denominator. - Both numerator and denominator are degree 1. - The horizontal asymptote is the ratio of the leading coefficients: \( y = \frac{-11}{1} = -11 \) **Summary:** - **Domain:** \( x \neq 0 \) - **Vertical Asymptote:** \( x = 0 \) - **Horizontal Asymptote:** \( y = -11 \) --- ### 2. \( \frac{7}{x-4} \) **Domain:** - The denominator \( x - 4 \) cannot be zero. - **Domain:** \( x \neq 4 \) **Vertical Asymptote:** - Occurs where the denominator is zero. - \( x = 4 \) **Horizontal Asymptote:** - The degrees of the numerator and denominator are both 1. - The horizontal asymptote is the ratio of the leading coefficients: \( y = \frac{7}{1} = 7 \) **Summary:** - **Domain:** \( x \neq 4 \) - **Vertical Asymptote:** \( x = 4 \) - **Horizontal Asymptote:** \( y = 7 \) --- ### 3. \( \frac{x}{x^{2}-16} \) First, factor the denominator: \( x^{2} - 16 = (x - 4)(x + 4) \) So, the function becomes: \( \frac{x}{(x - 4)(x + 4)} \) **Domain:** - The denominator \( (x - 4)(x + 4) \) cannot be zero. - \( x \neq 4 \) and \( x \neq -4 \) - **Domain:** \( x \in \mathbb{R} \), \( x \neq \pm 4 \) **Vertical Asymptotes:** - Occur where the denominator is zero. - \( x = 4 \) and \( x = -4 \) **Horizontal Asymptote:** - Compare the degrees of the numerator and denominator. - Numerator degree: 1 - Denominator degree: 2 - Since the denominator has a higher degree, the horizontal asymptote is \( y = 0 \) **Summary:** - **Domain:** \( x \neq \pm 4 \) - **Vertical Asymptotes:** \( x = 4 \) and \( x = -4 \) - **Horizontal Asymptote:** \( y = 0 \) --- ### Visual Representation To better understand these functions, graphing them can be helpful. Here's a brief description of their graphs: 1. **\( \frac{-11}{x} \):** - Hyperbola with branches in the second and fourth quadrants. - Approaches \( x = 0 \) vertically and \( y = -11 \) horizontally. 2. **\( \frac{7}{x-4} \):** - Hyperbola shifted right by 4 units. - Approaches \( x = 4 \) vertically and \( y = 7 \) horizontally. 3. **\( \frac{x}{x^{2}-16} \):** - More complex hyperbola with vertical asymptotes at \( x = 4 \) and \( x = -4 \). - Approaches \( y = 0 \) horizontally. --- ### Additional Considerations - **Intercepts:** - **Y-Intercept:** Set \( x = 0 \) and solve for \( y \) (if within the domain). - **X-Intercept:** Set \( y = 0 \) and solve for \( x \). - **Behavior Near Asymptotes:** - Analyze the limits as \( x \) approaches the vertical asymptotes from the left and right. - **Symmetry:** - Determine if the functions are even, odd, or neither. If you have a specific question or need further analysis on these functions, feel free to ask!