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.