QUESTION \( 6: \) NORTH WIST Given: \( f(x)=\frac{2}{x-1}-2 \) 4.1 Write down the equations of the asymptotes of \( f \). (2) 4.2 Draw a graph of \( f \). Clearly label \( A L L \) the intercepts with the axes and the (4) asymptotes on your graph. 4.3 Determine the equation of the line of symmetry of \( f \) for \( m<0 \). 4.4 For which values of \( x \) will \( f(x) \leq-4 \) ? [10]

To set the stage for exploring the function \( f(x) = \frac{2}{x-1} - 2 \), remember that asymptotes give us vital clues about the behavior of rational functions. In this case, the vertical asymptote occurs where the denominator is zero, i.e., when \( x=1 \). The horizontal asymptote for rational functions can typically be identified as the value that \( f(x) \) approaches as \( x \) goes to infinity. Here, as \( x \) tends to infinity, \( f(x) \) approaches -2, leading to a horizontal asymptote at \( y=-2 \). When graphing \( f(x) \), start by plotting the vertical asymptote along \( x=1 \) and the horizontal asymptote along \( y=-2 \). The x-intercept can be found by setting \( f(x) = 0 \) and solving, yielding an x-intercept at \( x=3 \). The y-intercept is discovered by substituting \( x=0 \), resulting in \( f(0)=-2 \). Remember to label these points clearly on your graph, ensuring a visual reference for the function's behavior around its asymptotes and intercepts!