iv. sketch the graph of each of the following functions. a f(x)=(1)/(x-4) b g(x)=(x+2)/(x^2)+5x+6 C h(x)=(x^2+1)/(x^2)-4x+4 d, k(x)=(x+6)/(x^2)+4 Let The ratio of boys to girls in a certain high school

To sketch the graphs of the given functions, we analyze their properties such as domain, asymptotes, intercepts, and behavior. Here's a breakdown for each function:<br /><br />---<br /><br />### **a) \( f(x) = \frac{1}{x-4} \)**<br /><br />- **Domain**: \( x \neq 4 \) (denominator cannot be zero).<br />- **Vertical Asymptote**: \( x = 4 \) (where the denominator becomes zero).<br />- **Horizontal Asymptote**: \( y = 0 \) (as \( x \to \pm\infty \), \( f(x) \to 0 \)).<br />- **Behavior**: The graph is a hyperbola with branches in quadrants I and III, centered around the vertical asymptote \( x = 4 \).<br /><br />---<br /><br />### **b) \( g(x) = \frac{x+2}{x^2 + 5x + 6} \)**<br /><br />- Factorize the denominator: \( x^2 + 5x + 6 = (x+2)(x+3) \).<br />- **Domain**: \( x \neq -2, -3 \) (denominator cannot be zero).<br />- **Vertical Asymptotes**: \( x = -2 \) and \( x = -3 \).<br />- **Simplification**: The numerator \( x+2 \) cancels with one factor in the denominator, leaving \( g(x) = \frac{1}{x+3} \).<br />- **Horizontal Asymptote**: \( y = 0 \) (as \( x \to \pm\infty \), \( g(x) \to 0 \)).<br />- **Behavior**: The graph resembles a shifted hyperbola with a vertical asymptote at \( x = -3 \).<br /><br />---<br /><br />### **c) \( h(x) = \frac{x^2 + 1}{x^2 - 4x + 4} \)**<br /><br />- Factorize the denominator: \( x^2 - 4x + 4 = (x-2)^2 \).<br />- **Domain**: \( x \neq 2 \) (denominator cannot be zero).<br />- **Vertical Asymptote**: \( x = 2 \).<br />- **Horizontal Asymptote**: Since the degrees of the numerator and denominator are equal, divide the leading coefficients: \( y = \frac{1}{1} = 1 \).<br />- **Behavior**: The graph approaches \( y = 1 \) as \( x \to \pm\infty \), with a vertical asymptote at \( x = 2 \).<br /><br />---<br /><br />### **d) \( k(x) = \frac{x+6}{x^2 + 4} \)**<br /><br />- **Domain**: All real numbers (\( x^2 + 4 > 0 \) for all \( x \)).<br />- **Vertical Asymptote**: None (denominator never equals zero).<br />- **Horizontal Asymptote**: Since the degree of the numerator is less than the degree of the denominator, \( y = 0 \).<br />- **Behavior**: The graph has no vertical asymptotes and approaches \( y = 0 \) as \( x \to \pm\infty \).<br /><br />---<br /><br />For the second part of your question regarding the ratio of boys to girls in a high school, please provide more details or clarify the context so I can assist further!