What is the domain of the function $y=\ln (x+2)$ ?
The function provided is \(y = \ln(x + 2)\). The natural logarithm function, denoted as \(\ln(u)\), is defined if and only if its argument \(u\) is strictly greater than zero, i.e., \(u > 0\). In this specific function, the argument is the expression \(x + 2\). To find the domain of the function, we must identify all values of \(x\) for which this expression is positive.
To determine the domain, we set the argument of the logarithm to be greater than zero:
\[x + 2 > 0\]
To solve for \(x\), we subtract 2 from both sides of the inequality:
\[x > -2\]
This inequality defines all values of \(x\) that are greater than \(-2\). Any value of \(x\) less than or equal to \(-2\) would make the argument of the logarithm non-positive, rendering the function undefined in the set of real numbers.
Based on our calculation, the domain of the function \(y = \ln(x + 2)\) consists of all real numbers \(x\) such that \(x > -2\). Comparing this result to the given options, we find that the correct choice is \(x > -2\).
Understanding the domain of a function is crucial in real-world modeling, such as in finance or population growth models where variables like time or investment amount cannot be negative or zero. For instance, if you are modeling the growth of an investment over time \(t\) using a logarithmic scale, knowing the domain helps ensure that your model is only applied to valid time intervals, preventing mathematically impossible calculations like taking the logarithm of a negative number. This fundamental skill allows for the creation of reliable and meaningful scientific models.