Let's break down the function and graph it step by step.

The given function is:

$x = -\log^2(x)$

This equation is a bit unconventional in its form. The notation $\log^2(x)$ typically means $(\log(x))^2$. So, the equation is really:

$x = -(\log(x))^2$

Step 1: Asymptote

For the logarithmic function $\log(x)$, there is an asymptote at $x = 0$ because the logarithmic function is undefined for non-positive values of $x$. Since $(\log(x))^2$ is still dependent on $\log(x)$, the function is also undefined at $x \leq 0$, so the vertical asymptote is at $x = 0$.

Step 2: Plotting Points

To plot points, we need to evaluate the function at specific positive values of $x$. Since $\log(x)$ is only defined for positive $x$, let’s choose a few values for $x$.

For $x = 1$:

$y = -(\log(1))^2 = -(0)^2 = 0$ So, one point is $(1, 0)$.

For $x = 10$:

$y = -(\log(10))^2 = -(1)^2 = -1$ So, another point is $(10, -1)$.

Step 3: Graphing the function

• The graph of $y = -(\log(x))^2$ will be a curve that starts near $x = 0$ and gradually slopes downward as $x$ increases.
• The function will be concave down (the negative square of a logarithm means the graph will open downwards).
• There is a vertical asymptote at $x = 0$.

Domain and Range

• Domain: The function is only defined for $x > 0$, so the domain is: $\text{Domain: } (0, \infty)$

• Range: Since the logarithmic square is always non-negative, and we negate it, the function values can never be positive. As $x \to \infty$, $y \to -\infty$. So, the range is: $\text{Range: } (-\infty, 0]$

Graph Interpretation

1. The graph will approach $-\infty$ as $x$ increases.
2. The asymptote at $x = 0$ will make the curve approach from the right.

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Would you like me to create the visual graph for you? Or do you need further clarification on any of the steps?

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Here are some related questions to deepen your understanding:

1. How does the graph of $y = \log(x)$ behave, and how does it relate to this transformation?
2. How would the graph change if the equation were $y = -\log^2(x + 2)$?
3. What happens if the base of the logarithm changes to something other than 10 (e.g., base 2)?
4. How does the graph of a logarithmic function differ from an exponential function?
5. What is the effect of adding a constant to a logarithmic function like $y = -(\log(x))^2 + 3$?

Tip: When working with logarithmic transformations, remember that any transformation inside the logarithmic function (like shifting or stretching) will affect the horizontal aspect of the graph, while transformations outside will affect the vertical.