QUESMON 6 NORTH HEST Given: \( f(x)=\frac{2}{x-1}-2 \) (2) 4.1 Write down the equations of the asymptotes of \( f \). 4.2 Draw a graph of \( f \). Clearly label ALL 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 \( m<0 \). (2) 4.4 For which values of \( x \) will \( f(x) \leq-4 \) ? [10]
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The function provided is \( f(x) = \frac{2}{x-1} - 2 \). To analyze the function, let’s dive into asymptotes and graph: **4.1** Asymptotes: The vertical asymptote occurs when the denominator is zero. Thus, \( x - 1 = 0 \) gives \( x = 1 \). For horizontal asymptotes, we can see that as \( x \to \pm \infty \), the \( \frac{2}{x-1} \) term approaches 0, so the horizontal asymptote is \( y = -2 \). **4.2** Graph of \( f \): When drawing the graph, plot the vertical line at \( x = 1 \) and the horizontal line at \( y = -2 \). The intercepts can be found by setting \( x = 0 \) to find the y-intercept, \( f(0) = \frac{2}{-1} - 2 = -3 \), giving a point \( (0, -3) \). For x-intercept, set \( f(x) = 0 \) which leads to \( x = 3 \). Mark points \( (0, -3) \) and \( (3, 0) \) on your graph. **4.3** Line of Symmetry: The function can be transformed to find symmetry. For \( m < 0 \), the line of symmetry is given as \( x = \frac{1}{2} \), located halfway between the asymptote at \( x=1 \) and the intercept at \( x=3 \). **4.4** Solve \( f(x) \leq -4 \): Set up the inequality: \[ \frac{2}{x-1} - 2 \leq -4 \] By simplifying, you arrive at \( \frac{2}{x-1} \leq -2 \) which leads to \( 2 \leq -2(x-1) \), or, \( x < 0 \) and \( x > 3 \). Therefore, \( f(x) \leq -4 \) for all \( x < 0 \) or \( x > 3 \). With this analysis, you have a thorough understanding of the function, its behavior, and the requested aspects!
