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Understanding the domain of a natural logarithm function is crucial for graphing and solving problems involving logarithms. The natural logarithm, denoted as \(\ln(x)\), is defined only for positive values of \(x\). This means the input to the \(\ln\) function must be greater than zero (
\(x>0\)).

For the given function \(g(x) = \ln(-x)\), we must consider when \(\-x>0\), which translates to all the values for which \(x<0\). Consequently, the domain of \(g(x)\) is the set of all negative real numbers. In simpler terms, you can only plug negative numbers into \(g(x)\) to get a real result. It's important to remember that natural logarithms do not accept zero or negative numbers as inputs, which aligns with the logarithmic property that the argument must be positive.

The \(x\)-intercept of a function is the point at which the graph of the function crosses the \(x\)-axis. To find the \(x\)-intercept, we set the function equal to zero and solve for \(x\).

For \(g(x) = \ln(-x)\), we calculate the \(x\)-intercept by solving \(\ln(-x) = 0\). Since the exponential form of the natural logarithm has base \(e\), we express this equation as \(e^{\ln(-x)} = e^0\), which simplifies to \(\-x = 1\). By multiplying both sides by \(\-1\), we find that the \(x\)-intercept is \(x = -1\). This tells us that the graph of our function touches the \(x\)-axis exactly at the point \(\-1, 0\). Knowing the \(x\)-intercept is useful not only for graphing but also for understanding the behavior of the function in relation to the \(x\)-axis.

A vertical asymptote is a line that the graph of a function approaches but never actually touches or crosses. It occurs at values of \(x\) where the function becomes undefined or approaches infinity.

For logarithmic functions, such as \(g(x) = \ln(-x)\), the vertical asymptote occurs where the argument of the logarithm is zero since the logarithm of zero is undefined. In our exercise, the argument \(\-x\) becomes zero when \(x = 0\). Therefore, the vertical asymptote for the function is the line \(x = 0\). This is a critical feature of the graph representing an 'invisible barrier' that the function values get infinitely close to but cannot cross. When graphing \(g(x)\), remember that the graph will approach this vertical asymptote from the left side, as the domain consists of negative \(x\)-values only.

Graphing logarithmic functions like \(g(x) = \ln(-x)\) involves understanding the function's domain, \(x\)-intercept, and asymptotes. The domain informs us which \(x\)-values are allowed, the \(x\)-intercept indicates where the graph intersects the \(x\)-axis, and the vertical asymptote shows where the function tends toward negative or positive infinity.

When graphing \(g(x)\), start by plotting the \(x\)-intercept, which we've determined to be \(x = -1\). Next, draw the vertical asymptote as a dashed line along the \(x = 0\) line. Since the function's domain is \(x<0\), we'll only sketch the graph on the left side of the \(y\)-axis. The curve approaches the \(x\)-axis near the \(x\)-intercept and then sweeps downward, getting closer and closer to the vertical asymptote without crossing it.

It's worth noting that \(g(x) = \ln(-x)\) is a reflection of \(\ln(x)\) across the \(y\)-axis due to the negative sign within the logarithm. This reflection results in a graph that decays towards the left, or negative side of the graph, illustrating that mathematical transformations can significantly change the function's graphical representation.