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Understanding the domain of a function is crucial for sketching its graph. The domain refers to the set of all possible input values (typically represented by 'x' for real functions) for which the function is defined. In the case of logarithmic functions, like \(h(x)=\ln (x+2)\), the key is to ensure the argument of the logarithm is greater than zero, because the logarithm is undefined for nonpositive values.

For \(h(x)=\ln (x+2)\), the input \(x+2\) must be positive; therefore, we arrive at the inequality \(x > -2\). This gives us an open interval for the domain: \( (-2, +\infty) \), meaning that the input values can range from just above -2 to positive infinity. Not including -2 is important because the point at which the argument of the logarithmic function reaches zero is not included in the domain.

When graphing or tackling problems involving the domain of logarithmic functions, always look for these critical points—values which make the argument zero or negative—since these are the boundaries of your domain. The proper identification of domain affects all other aspects of understanding functions, from solving equations to interpreting graphs.

Inequalities involving logarithms often seem daunting, but they follow logical steps that can be simplified through understanding. A logarithmic inequality, such as \(x + 2 > 0\), is derived from the requirement that a logarithmic function's input must be greater than zero.

To solve the inequality \(x + 2 > 0\), we simply isolate the variable \(x\), resulting in \(x > -2\). This translates to the range of permissible 'x' values that make \(h(x)=\ln (x+2)\) defined. Whenever you face a logarithmic inequality, your goal is to find the interval of x-values that will keep the log's argument positive.

Solving logarithmic inequalities is imperative when working with logarithmic functions, as it ensures the arguments fall within the domain, enabling the function to have real number outputs.

Tip: Always remember to check if there’s a need to flip the inequality sign when both sides of the inequality are multiplied or divided by a negative number. This isn't the case for the inequality \(x + 2 > 0\) because it remains straightforward.

The interpretation of graphs is vital for visualizing relationships between variables. When it comes to the logarithmic function \(h(x)=\ln (x+2)\), the graph helps to show how \(y\) values change with \(x\). As we observe the plotted points from the exercise, the graph reveals several features.

A logarithmic curve is generally characterized by a sharp increase to the left of the y-axis and a slower, flattening growth to the right. For the function at hand, we note that the curve approaches the vertical line \(x=-2\) but never touches or crosses it, reflecting the domain's left-boundary. As \(x\) increases, the function increases as well but at a decreasing rate.

This observed increase in the value of \(h(x)\) as \(x\) increases indicates that the function is increasing on its domain. Furthermore, the graph does not intersect the y-axis, which is consistent with the fact that the logarithm of 1 (\(\ln(1)=0\)) is a distinctive property of log functions.

Tip: Use the graph to confirm properties like the domain and to visualize the function's behavior, such as rates of increase or decrease, asymptotic behavior (approaching a line without ever touching it), and key points that provide insight into the function's characteristics.