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When we talk about the domain of a function, we are essentially asking: "What values can we input into this function?" For the natural logarithmic function presented, namely \( f(x) = \ln(x^2 + 1) \), we need to ensure that anything under the logarithm sign is positive.

Here's the good news: the expression inside the logarithm, \( x^2 + 1 \), is always positive no matter what real number you choose for \( x \). This is because squaring any real number, \( x^2 \), yields a non-negative result, and adding 1 makes it strictly positive.

Thus, there are no restrictions on \( x \) and it can take any real value. The domain is therefore all real numbers, expressed in interval notation as \( (-\infty, \infty) \). Here’s a summary of the key points:

• For the function \( f(x) = \ln(x^2 + 1) \), the argument \( x^2 + 1 \) is positive for every real \( x \).
• This means \( x \) has no restrictions, leading to a domain of all real numbers.
• In interval notation, we express this as \((-\infty, \infty)\).

Graphing a function helps us visually comprehend how the function behaves across its domain. For the natural logarithmic function \( f(x) = \ln(x^2 + 1) \), there are some key aspects to consider when sketching its graph.

Firstly, as \( x \) approaches 0, the expression\( x^2 + 1 \) equals 1, thus \( f(x) \) approaches \( \ln(1) \) which is 0. This means the graph touches the y-axis at the point\((0, 0)\). Secondly, as the absolute value of \( x \) becomes large, \( x^2 + 1 \) increases, resulting in \( f(x) \) rising slowly.

The graph is smooth and continuous, increasing without bound as \( |x| \) becomes very large. Key takeaways for graphing this function:

• The graph passes through the origin at \((0, 0)\).
• As \( x \) moves away from zero in either direction, \( x^2 + 1 \) grows, causing \( f(x) \) to slowly rise.
• The increase is gradual, reflecting the logarithmic nature of the function.

Mathematical continuity describes a function's unbroken, smooth curve without gaps, jumps, or visible interruptions. For the function \( f(x) = \ln(x^2 + 1) \), understanding its continuity gives us insights into its behavior.

Since the expression \( x^2 + 1 \) is positive for all real \( x \), this ensures that the function \( \ln(x^2 + 1) \) is defined everywhere in its domain. Consequently, there are no points of discontinuity or breaks, meaning that the graph is a continuous curve. The smooth and gradual increase of the graph further emphasizes its continuity. Here's what to note about this function's continuity:

• The function is continuous over all real numbers, thanks to \( x^2 + 1 > 0 \) for all real \( x \).
• There are no jumps, gaps, or asymptotes in the graph; it flows naturally.
• As a natural logarithmic function with a positive argument, it remains well-behaved.

Therefore, for students exploring mathematical continuity, \( f(x) = \ln(x^2 + 1) \) offers a perfect example of a function that exhibits smooth continuity over its entire domain.