To graph the function $y = \frac{1}{\ln(x)} - 2$, you can follow these steps:

1. Understand the function's components:

• The natural logarithm, $\ln(x)$, is defined only for $x > 0$. So, the domain of the function is $x > 0$.
• The term $\frac{1}{\ln(x)}$ will be undefined when $\ln(x) = 0$. Since $\ln(x) = 0$ when $x = 1$, there is a vertical asymptote at $x = 1$.
• The function includes a transformation: the $-2$ term shifts the entire graph downward by 2 units.

2. Find the asymptotes:

• Vertical asymptote: As noted earlier, $\ln(x) = 0$ at $x = 1$, so there is a vertical asymptote at $x = 1$.
• Horizontal asymptote: As $x \to \infty$, $\ln(x) \to \infty$, so $\frac{1}{\ln(x)} \to 0$. Therefore, the function approaches $y = -2$ as $x$ increases. This is the horizontal asymptote.

3. Plot key points:

You can find some key points by choosing values of $x$ greater than 0 and calculating $y$. For example:

• At $x = 2$, $\ln(2) \approx 0.693$, so $y = \frac{1}{0.693} - 2 \approx 1.442 - 2 = -0.558$.
• At $x = 3$, $\ln(3) \approx 1.099$, so $y = \frac{1}{1.099} - 2 \approx 0.910 - 2 = -1.090$.
• At $x = 10$, $\ln(10) \approx 2.302$, so $y = \frac{1}{2.302} - 2 \approx 0.434 - 2 = -1.566$.

4. Sketch the graph:

• Start by plotting the vertical asymptote at $x = 1$.
• Draw the horizontal asymptote at $y = -2$.
• Plot the points you calculated (e.g., for $x = 2, 3, 10$).
• Connect the points smoothly, ensuring the graph approaches the asymptotes.

5. Shape of the graph:

• As $x$ approaches 1 from the right, the function goes to $+\infty$ because $\ln(x)$ becomes small.
• As $x$ moves to large values, the graph will approach $y = -2$ from above, leveling off.

6. Key characteristics of the graph:

• Domain: $x > 0$.
• Vertical asymptote: At $x = 1$.
• Horizontal asymptote: At $y = -2$.
• Behavior near $x = 1$: The graph increases steeply as $x$ approaches 1 from the right.
• Behavior as $x \to \infty$: The graph flattens out at $y = -2$.

Graphing Tip:

Always begin with identifying asymptotes and key points, then plot them before connecting the dots. This helps you maintain the overall shape and ensure accuracy.

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Would you like me to explain any of these steps in more detail? Or help with additional graphing questions?

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Related Questions:

1. How does the graph change if the function is $y = \frac{1}{\ln(x)} + 3$?
2. How would the graph look for $y = \frac{1}{\ln(x - 1)} - 2$?
3. What is the impact of a horizontal shift on this function?
4. How do we find the range of the function $y = \frac{1}{\ln(x)} - 2$?
5. How do we determine where the function is increasing or decreasing?

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Tip: When graphing a function with a natural logarithm, always check for vertical asymptotes and domain restrictions first. This will help you identify regions where the function is undefined and avoid drawing incorrect portions of the graph.