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.

Discover how to find the domain and range of the exponential function y = e^{-x} - 2. This problem covers key concepts in exponential functions, including determining the domain as (-∞, ∞) and the range as (-2, ∞). Suitable for high school students in Grades 10-12.

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