The graph of the function f(x) = log₁ₑ(x), also written as ln(x), represents the natural logarithm of x. In this response, we will discuss the graph's domain, range, and three important properties.

Domain:
The domain of the natural logarithm function f(x) = ln(x) is the set of all positive real numbers, excluding zero. This is because the natural logarithm is only defined for positive values. Therefore, the domain can be expressed as D = (0, ∞).

Range:
The range of the natural logarithm function f(x) = ln(x) is the set of all real numbers. In other words, the function can take any real value as its output. Therefore, the range can be expressed as R = (-∞, ∞).

Properties:
1. Increasing Function: The natural logarithm function is an increasing function. This means that as x increases, the corresponding values of ln(x) also increase. This property can be observed by analyzing the graph. As x moves towards positive infinity, ln(x) approaches infinity. Similarly, as x approaches zero, ln(x) tends towards negative infinity.

2. Asymptote: The graph of f(x) = ln(x) has a vertical asymptote at x = 0. This means that the graph approaches the y-axis but never touches it. As x approaches zero from the right side, ln(x) tends towards negative infinity. On the other hand, as x approaches zero from the left side, ln(x) tends towards positive infinity. This behavior is indicated by the graph as it becomes steeper and steeper as x approaches zero.

3. One-to-One Function: The natural logarithm function is a one-to-one function, which means that each input value corresponds to a unique output value. This property can be observed by analyzing the graph. As the graph moves from left to right, it never intersects or overlaps itself, indicating the uniqueness of the function.

Graph:
The graph of f(x) = ln(x) starts from the point (1, 0) and gradually increases as x increases. It approaches the y-axis but never touches it, indicating the vertical asymptote at x = 0. The graph becomes steeper and steeper as x approaches zero and approaches positive infinity as x increases. The graph is continuous and smooth without any breaks or sharp turns.

In conclusion, the graph of f(x) = ln(x) has a domain of (0, ∞), a range of (-∞, ∞), and exhibits important properties such as being an increasing function, having a vertical asymptote at x = 0, and being a one-to-one function.