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The domain of a function is crucial because it tells us the set of all possible input values (usually designated as 'x') that will produce a valid output in the function. For the function \(g(x) = \ln(-x)\), we need to determine which values of \(x\) can be used to compute \(g(x)\) without causing a mathematical error.

The natural logarithm, \(\ln(x)\), is only defined for positive values of \(x\). Therefore, we must ensure that the input for the natural logarithm part, \(-x\), is positive. To find when \(-x\) is positive, we set up the inequality \(-x > 0\). Solving \(-x > 0\) involves dividing both sides by -1, which also flips the inequality sign, giving us \(x < 0\). Thus, the domain of our function is all negative real numbers, which can be expressed in interval notation as \(( -\infty, 0)\).

Understanding the domain helps in determining where our function can be graphed and evaluated. It prevents undefined mathematical operations, especially in logarithmic or rational functions.

Graphing functions involves plotting the equation on a coordinate system to visualize its behavior. For \(g(x) = \ln(-x)\), graphing it first requires knowing its domain. Since we have already established that x is less than 0, the graph of this function will reside only in the second quadrant of the Cartesian plane.

The function \(g(x)\) approaches \(\infty\) as \(x\) approaches 0 from the left. This graph resembles a reflection of the standard logarithmic function \(\ln(x)\) across the y-axis, due to the negative sign. Here are some features to note while graphing:

• The graph never crosses the y-axis because \(x\) can never be 0.
• It rises towards infinity as \(x\) gets closer to 0 from the left.
• It is undefined for \(x \geq 0\), hence no part of the graph appears in the first quadrant.

This visualization helps in understanding how the function behaves and ensures that the derived domain is respected.

Inequalities are statements about the relative size of two values, often involving expressions like greater than (\(>\)) or less than (\(<\)). In mathematics, solving inequalities is essential when determining where a function is defined or analyzing its behavior.

In the case of \(g(x) = \ln (-x)\), inequalities help us find the domain. We used the inequality \(-x > 0\) to determine that \(x\) must be less than 0 for the function to be defined. Understanding how to manipulate inequalities is key:

• Simplify complex expressions to create a straightforward inequality.
• Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.
• Apply the solution to real-world contexts like function domains or ranges.

These steps are fundamental in defining where functions "live" and can be particularly useful in higher-level mathematics and real-world applications.