13 Consider: \( g(x)=\frac{a}{x+p}+q \) The following information of \( g \) is given: • Domain: \( x \in R ; x \neq-2 \) - \( x \)-intercept at \( \mathrm{K}(1 ; 0) \) - \( y \)-intercept at \( \mathrm{N}\left(0 ;-\frac{1}{2}\right) \)

To solve for the parameters \( a \), \( p \), and \( q \) in the function \( g(x) = \frac{a}{x+p} + q \) given the conditions, we will follow these steps: 1. **Identify the known conditions:** - The domain of \( g \) is \( x \in \mathbb{R} \) and \( x \neq -2 \), which implies \( p = -2 \) (since the function is undefined at \( x = -p \)). - The \( x \)-intercept is at \( K(1, 0) \), meaning \( g(1) = 0 \). - The \( y \)-intercept is at \( N(0, -\frac{1}{2}) \), meaning \( g(0) = -\frac{1}{2} \). 2. **Substituting the intercepts into the function:** - For the \( x \)-intercept: \[ g(1) = \frac{a}{1 - 2} + q = 0 \] This simplifies to: \[ \frac{a}{-1} + q = 0 \implies -a + q = 0 \implies q = a \] - For the \( y \)-intercept: \[ g(0) = \frac{a}{0 - 2} + q = -\frac{1}{2} \] This simplifies to: \[ \frac{a}{-2} + q = -\frac{1}{2} \] Substituting \( q = a \) into this equation gives: \[ \frac{a}{-2} + a = -\frac{1}{2} \] To combine the terms, we can express \( a \) as: \[ -\frac{a}{2} + a = -\frac{1}{2} \implies \frac{a}{2} = -\frac{1}{2} \implies a = -1 \] 3. **Finding \( q \):** Since \( q = a \), we have: \[ q = -1 \] 4. **Summarizing the results:** - \( p = -2 \) - \( a = -1 \) - \( q = -1 \) Thus, the parameters are: - \( a = -1 \) - \( p = -2 \) - \( q = -1 \) The function can now be expressed as: \[ g(x) = \frac{-1}{x - 2} - 1 \]