Let's find the domain and range of the function $y = e^{-x} - 2$.

1. Domain:

The domain of a function consists of all the possible input values ($x$) for which the function is defined.

• The expression $e^{-x}$ (an exponential function) is defined for all real numbers $x$.
• Therefore, there are no restrictions on $x$.

Thus, the domain of $y = e^{-x} - 2$ is:

$\boxed{(-\infty, \infty)}$

2. Range:

The range consists of all the possible output values ($y$) that the function can produce.

• The function $e^{-x}$ is always positive for any real number $x$, and $e^{-x}$ can approach 0 as $x$ tends to infinity, but it never reaches 0. Therefore, $e^{-x} > 0$ for all $x$.
• Now, $y = e^{-x} - 2$ subtracts 2 from $e^{-x}$, meaning:
  • The minimum value of $e^{-x}$ approaches 0 as $x$ becomes very large, so $y$ approaches $-2$.
  • The maximum value of $e^{-x}$ is 1 when $x = 0$, so $y = 1 - 2 = -1$.

Thus, the function will never reach $-2$ but can get infinitely close to it, and it can take values less than $-1$.

Therefore, the range of the function is:

$\boxed{(-2, \infty)}$

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Would you like more details or have any questions on this? Here are 5 related questions to expand on this:

1. How would the domain and range change if the function was $y = e^{-x} + 2$?
2. What is the graph of the function $y = e^{-x} - 2$ like, and how does it behave as $x \to \infty$ and $x \to -\infty$?
3. Can you determine the horizontal asymptote for this function?
4. What would be the domain and range of $y = -e^{-x} - 2$?
5. How would the function change if the exponent in $e^{-x}$ were positive, like in $e^x - 2$?

Tip: Exponential functions always have horizontal asymptotes, which can help determine their range.