The range of the function y = 1 – 2ex is which of the
following?
Note: The interval [a, b] include the end-points a and b.
The interval (a, b) excludes the end-points a and b.
A. (–,1] C. (1, )
B. (–,1) D. [1, )

We are given the function:

\[
y = 1 - 2e^x
\]

where \( e^x \) is the exponential function.

### Step-by-step analysis:
1. **Exponential function behavior**:
   - The function \( e^x \) is always positive and increases as \( x \) increases.
   - As \( x \to \infty \), \( e^x \to \infty \).
   - As \( x \to -\infty \), \( e^x \to 0 \).

2. **Transforming \( e^x \)**:
   - The expression \( 1 - 2e^x \) will subtract values from 1. 
   - When \( x \to -\infty \), \( e^x \to 0 \), so \( y \to 1 \).
   - When \( x \to \infty \), \( e^x \to \infty \), so \( y \to -\infty \).

### Range of the function:
- The maximum value of \( y \) is when \( e^x \) is smallest, which occurs as \( x \to -\infty \), giving \( y = 1 \).
- The function can decrease without bound, so as \( x \to \infty \), \( y \to -\infty \).

Thus, the range of the function is \( (-\infty, 1] \).

### Correct answer:
**A. (–∞, 1]**

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Would you like more details or have any questions about this?  
Here are 5 related questions:
1. What is the domain of the function \( y = 1 - 2e^x \)?
2. How does the derivative of \( y = 1 - 2e^x \) affect the graph?
3. Can an exponential function have negative values?
4. What transformations were applied to the basic \( e^x \) function to get \( y = 1 - 2e^x \)?
5. How would the range change if the function were \( y = 2e^x - 1 \)?

**Tip**: When determining the range of a function, focus on the limits as \( x \) approaches infinity and negative infinity.

The range of the function y = 1 – 2e^x is which of the following? A. (–∞,1] B. (–∞,1) C. (1, ∞) D. [1, ∞)

Discover how to find the range of the function y = 1 - 2e^x. This detailed explanation includes the behavior of exponential functions and their transformations. Ideal for students in Grades 10-12.

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