https://file.aiquickdraw.com/mathbot/file/1727739942073ve7dgu9k.jpg

The function given in the image is:

\[
f(x) = e^{x - 1} - 2
\]

This is an exponential function where \( e \) is the base of the natural logarithm, and the exponent is \( x - 1 \). The entire expression is then shifted down by 2 units.

Would you like more details about how to analyze or manipulate this function, such as finding its derivative, solving for \( x \), or finding its behavior as \( x \) approaches infinity?

Here are five related questions that expand upon this concept:

1. How do you find the derivative of \( f(x) = e^{x - 1} - 2 \)?
2. What is the y-intercept of the function \( f(x) = e^{x - 1} - 2 \)?
3. How would you solve \( f(x) = 0 \) for \( x \)?
4. What is the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \)?
5. How do translations affect the graph of \( f(x) = e^x \)?

**Tip:** For exponential functions like \( e^x \), remember that horizontal shifts affect the input of the exponent (e.g., \( x - 1 \) shifts the graph 1 unit to the right), while vertical shifts affect the entire function directly (e.g., subtracting 2 shifts the graph down by 2).

What is the function f(x) = e^(x-1) - 2 and how do you analyze it?

Explore how to analyze the exponential function f(x) = e^(x-1) - 2. This includes understanding function shifts, solving for x, and finding the derivative. Suitable for students in Grades 10-12.

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