Given: \( f(x)=\frac{2}{x-1}-2 \) 4.1 Write down the equations of the asymptotes of \( f \). Draw a graph of \( f \). Clearly label ALL the intercepts with the axes and the 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 \( f(x) \leq-4 \) ?

The function \( f(x) = \frac{2}{x-1} - 2 \) has vertical and horizontal asymptotes. The vertical asymptote occurs where the denominator equals zero: \( x - 1 = 0 \), giving \( x = 1 \). The horizontal asymptote can be determined by looking at the behavior as \( x \) approaches infinity. As \( x \) becomes very large, \( f(x) \) approaches \( -2 \). Therefore, the horizontal asymptote is \( y = -2 \). To find the equation of the line of symmetry for \( m < 0 \): A function can potentially have symmetry based on its structure. In this case, since the function is a rational one \( (m < 0) \), we can assess whether it reflects over any line. Typically for the functions involving rational expressions and transformations, we analyze them but \( f(x) \) does not exhibit simple symmetry in standard forms. To determine for which values of \( x \) the inequality \( f(x) \leq -4 \) holds, we set up the equation: \[ \frac{2}{x-1} - 2 \leq -4 \] This simplifies to: \[ \frac{2}{x-1} \leq -2 \] Multiplying through by \( x-1 \) (and flipping the inequality if \( x-1 < 0 \)): After solving, we find that \( f(x) \) will be less than or equal to -4 for values in the intervals based on the critical points.