Given: \( f(x)=\frac{s}{x-2}+1 \) 10.1 Write down the equations of the asymptotes of \( f \). 10.2 Determatie coordinates of \( B \), the \( x \)-intercept of \( f \). 10.3 Write down the domain of \( g \) If \( g(x)=f(x+1) \). 10.4 One of the axes of symmetry of \( f \) is an increasing function. Write down the equation of this axis of symmetry.

Let's solve the problem step by step. ### 10.1: Write down the equations of the asymptotes of \( f \). The function given is \[ f(x) = \frac{s}{x-2} + 1 \] **Vertical Asymptote:** A vertical asymptote occurs where the function is undefined, which happens when the denominator is zero. Setting the denominator to zero: \[ x - 2 = 0 \implies x = 2 \] Thus, the equation of the vertical asymptote is: \[ x = 2 \] **Horizontal Asymptote:** To find the horizontal asymptote, we look at the behavior of \( f(x) \) as \( x \) approaches infinity. The term \( \frac{s}{x-2} \) approaches 0 as \( x \) goes to infinity. Therefore, the horizontal asymptote is: \[ y = 1 \] ### Summary of Asymptotes: - Vertical Asymptote: \( x = 2 \) - Horizontal Asymptote: \( y = 1 \) --- ### 10.2: Determine coordinates of \( B \), the \( x \)-intercept of \( f \). The \( x \)-intercept occurs when \( f(x) = 0 \). Setting the function equal to zero: \[ 0 = \frac{s}{x-2} + 1 \] Rearranging gives: \[ \frac{s}{x-2} = -1 \] Multiplying both sides by \( x - 2 \) (noting that \( x \neq 2 \)): \[ s = - (x - 2) \] Solving for \( x \): \[ s = -x + 2 \implies x = 2 - s \] Thus, the coordinates of \( B \) are: \[ B(2 - s, 0) \] --- ### 10.3: Write down the domain of \( g \) if \( g(x) = f(x + 1) \). To find the domain of \( g(x) \), we need to determine where \( f(x + 1) \) is defined. Substituting \( x + 1 \) into the function: \[ g(x) = f(x + 1) = \frac{s}{(x + 1) - 2} + 1 = \frac{s}{x - 1} + 1 \] The function \( g(x) \) is undefined when the denominator is zero: \[ x - 1 = 0 \implies x = 1 \] Thus, the domain of \( g \) is all real numbers except \( x = 1 \): \[ \text{Domain of } g: (-\infty, 1) \cup (1, \infty) \] --- ### 10.4: One of the axes of symmetry of \( f \) is an increasing function. Write down the equation of this axis of symmetry. The function \( f(x) \) is a rational function, and its axis of symmetry can be found by examining the vertical line that bisects the function. For rational functions of the form \( \frac{s}{x - a} + b \), the axis of symmetry is typically the vertical line through the \( x \)-coordinate of the vertical asymptote. Since the vertical asymptote is at \( x = 2 \), the axis of symmetry is: \[ x = 2 \] However, since the problem states that one of the axes of symmetry is an increasing function, we can also consider the symmetry about the horizontal asymptote. The function is symmetric about the line \( y = 1 \) as well. Thus, the equation of the axis of symmetry that is an increasing function is: \[ y = 1 \] ### Summary of Answers: - 10.1: Vertical Asymptote: \( x = 2 \), Horizontal Asymptote: \( y = 1 \) - 10.2: Coordinates of \( B \): \( (2 - s, 0) \) - 10.3: Domain of \( g \): \( (-\infty, 1) \cup (1, \infty) \) - 10.4: Equation of the axis of symmetry: \( y = 1 \)