The given function is:

$g(x) = e^{x+2} - 1$

Key Insights about this function:

1. Exponential Form: The expression $e^{x+2}$ is an exponential function. The base $e$ is approximately 2.718, and it grows exponentially as $x$ increases.
2. Translation: The function has been shifted horizontally by 2 units to the left due to the term $x + 2$ inside the exponent. Additionally, the subtraction of 1 shifts the graph downward by 1 unit.

Would you like help with any specific operations related to this function, such as finding its derivative, integral, or evaluating it at specific points?

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Related Questions:

1. What is the derivative of $g(x)$?
2. What happens to the function $g(x)$ as $x \to \infty$?
3. How do you find the inverse of the function $g(x)$?
4. How would you graph $g(x)$ using key points and transformations?
5. How do you integrate this function $g(x)$?

Tip: When dealing with exponential functions, remember that transformations such as shifts inside the exponent affect the graph in a way similar to how they would affect polynomials, but the growth remains exponentially fast.