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The domain of a function refers to the set of all possible input values for which the function is defined. Understanding this for the function \( y = \ln(x-3) \) is crucial before any graph sketching. The natural logarithm function, \( \ln(x) \), is only defined for positive values. For our function, you should solve \( x-3 > 0 \) to find where it is defined. This leads us to \( x > 3 \). Therefore, the domain of this function is \( (3, \infty) \).

This means the function doesn't exist at \( x = 3 \) but starts immediately after it. When defining the domain:

• The interval \( (3, \infty) \) indicates that as \( x \) starts just after 3, going all the way to infinity, the function is valid.
• This can be visualized on a number line as everything to the right of 3, excluding 3 itself.

Grasping the domain helps in determining key features in graph sketching and ensures that you're solving within the function’s valid perimeter.

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \). The number \( e \) (approximately 2.718) is a fundamental constant in mathematics, similar to the value of \( \pi \) for circles. The natural logarithm possesses unique properties:

• \( \ln(1) = 0 \)
• \( \ln(e) = 1 \)

These properties assist in plotting key points when graphing, such as when \( x=4 \), thus making \( \ln(4-3)=0 \).

The behavior of \( \ln(x-3) \) can be further analyzed:

• As \( x \) gets very close to 3, \( x-3 \) approaches zero from the positive side, meaning \( y = \ln(x-3) \to -\infty \). This illustrates how the graph swoops deeply as it nears the vertical line \( x=3 \).
• As \( x \) increases further beyond any limit, \( x-3 \) grows larger, leading \( y = \ln(x-3) \to \infty \). This depicts the ever-rising trend of the graph.

Understanding these behaviors emphasizes important trends useful in graph sketching and analysis.

Asymptotes are crucial lines that a graph approaches but never touches or crosses. In the graph of \( y = \ln(x-3) \), a vertical asymptote is present at \( x=3 \).

Here's what this means:

• As \( x \) approaches 3 from the right (but never actually reaches or crosses 3), the value of \( y \) sharply drops to \(-\infty\).
• This vertical asymptote acts like a boundary that the function will always strive towards indefinitely without actually meeting it. It implies that the graph will steeply rise or fall near this line.

Asymptotes guide the sketching of the graph, marking boundaries for its path as the curve nears the line. They significantly affect the visual layout and enhance the understanding of the graph’s behavior over its domain.