Sketch the graph of \( f(x)=\frac{k}{x+p}+q \) if - The domain is grven as \( x \in R ; x \neq-1 \) - The range is given as \( y \in R ; y \neq 2 \) - \( k<0 \) - \( x \)-intercept : \( \left(\frac{1}{2} ; 0\right) \) - \( f(0)=1 \)

To sketch the graph of \( f(x)=\frac{k}{x+p}+q \) given the specified conditions, we first need to analyze the function and how the parameters affect its shape. Since \( k < 0 \), the graph will be a hyperbola that opens downwards. The presence of \( p \) will shift the vertical asymptote, while \( q \) will determine the horizontal asymptote of the function. Since the range excludes \( y = 2 \), this means the horizontal asymptote of the function must be located at \( y = 2 \). Therefore, we set \( q = 2 \). Now, we can use the information about the \( x \)-intercept and \( f(0) \): 1. The \( x \)-intercept at \( \left(\frac{1}{2}, 0\right) \) gives: \[ 0 = \frac{k}{\frac{1}{2} + p} + q \Rightarrow \frac{k}{\frac{1}{2} + p} + 2 = 0 \Rightarrow \frac{k}{\frac{1}{2} + p} = -2 \] Hence, \( k = -2 \left(\frac{1}{2} + p\right) \). 2. At \( f(0) = 1 \): \[ 1 = \frac{k}{0 + p} + q \Rightarrow 1 = \frac{k}{p} + 2 \Rightarrow \frac{k}{p} = -1 \Rightarrow k = -p \] From the equation \( k = -2 \left(\frac{1}{2} + p\right) \) and \( k = -p \), we can set the equations equal to each other: \[ -p = -2\left(\frac{1}{2} + p\right) \Rightarrow p = 2 \left(\frac{1}{2} + p\right). \] Simplifying this leads to \[ p = 1 + 2p \Rightarrow -p = 1 \Rightarrow p = -1. \] Now substituting back to find \( k \): \[ k = -(-1) = 1. \] Substituting \( p \) and \( k \) into the function, we rewrite it as: \[ f(x) = \frac{-2}{x - 1} + 2 \] This function has a vertical asymptote at \( x = -1 \) and a horizontal asymptote at \( y = 2 \). To sketch the graph: - Mark the vertical asymptote at \( x = -1 \). - Mark the horizontal asymptote at \( y = 2 \). - At \( x = 0 \), the function value is \( 1 \). - The function crosses the x-axis at \( \frac{1}{2} \). The graph would be a hyperbola in the regions defined by these asymptotes, falling towards the horizontal asymptote from above in the left quadrant and below on the right side, while avoiding the vertical asymptote.