Sange of \( 3-\sqrt{e^{x+2}-e} \)

To understand the expression \( 3 - \sqrt{e^{x+2} - e} \), it's essential to recognize that it combines both constants and variable components. The term \( e^{x+2} \) can be rewritten as \( e^2 \cdot e^x \), and since \( e \) is approximately 2.718, this expression could represent a range of values depending on \( x \). Now, let's talk about when the expression under the square root, \( e^{x+2} - e \), is non-negative, as we need that to ensure the square root is real. For \( e^{x+2} - e \geq 0 \), we find that \( e^{x+2} \geq e \), which simplifies to \( e^x \geq e^{-2} \) or \( x \geq -2 \). Therefore, the expression is defined for \( x \geq -2 \). This can be quite useful in real-world applications, such as decay models, where understanding the threshold values where certain calculations become feasible is crucial. Imagine a scenario in finance where you're exploring the limits of a certain investment growth model.