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

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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 $$ f(x) \leq-4 $$ ?

UpStudy AI · Accepted Answer

It seems that I am unable to generate a plot at this moment. However, I can guide you on how to sketch the graph of $$ f(x) $$ based on the information we have gathered.

### Instructions to Sketch the Graph of $$ f(x) $$

1. **Draw the Axes**: Start by drawing the x-axis and y-axis.

2. **Plot the Asymptotes**:
   - Draw a dashed vertical line at $$ x = 1 $$ (vertical asymptote).
   - Draw a dashed horizontal line at $$ y = -2 $$ (horizontal asymptote).

3. **Plot the Intercepts**:
   - Mark the x-intercept at $$ (2, 0) $$.
   - Mark the y-intercept at $$ (0, -4) $$.

4. **Behavior of the Function**:
   - As $$ x $$ approaches $$ 1 $$ from the left, $$ f(x) $$ will approach $$ -\infty $$.
   - As $$ x $$ approaches $$ 1 $$ from the right, $$ f(x) $$ will approach $$ +\infty $$.
   - As $$ x $$ approaches $$ -\infty $$, $$ f(x) $$ approaches $$ -2 $$.
   - As $$ x $$ approaches $$ +\infty $$, $$ f(x) $$ also approaches $$ -2 $$.

5. **Sketch the Curve**:
   - Draw the curve approaching the vertical asymptote from below as it approaches $$ x = 1 $$ from the left.
   - Draw the curve approaching the vertical asymptote from above as it approaches $$ x = 1 $$ from the right.
   - Ensure the curve approaches the horizontal asymptote $$ y = -2 $$ as $$ x $$ goes to both $$ -\infty $$ and $$ +\infty $$.

6. **Label Everything**: Clearly label the asymptotes, intercepts, and any important points on the graph.

This will give you a complete sketch of the function $$ f(x) $$. If you have any further questions or need additional assistance, feel free to ask!
**Asymptotes:** - Vertical: $$ x = 1 $$ - Horizontal: $$ y = -2 $$ **Intercepts:** - x-intercept: $$ (2, 0) $$ - y-intercept: $$ (0, -4) $$ **Line of Symmetry for $$ m < 0 $$:** - $$ y = -2 $$ **Values of $$ x $$ where $$ f(x) \leq -4 $$:** - $$ 0 \leq x < 1 $$

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