The given function is:

$f(x) = e^x - 1 - 7$

To simplify, we can rewrite it as:

$f(x) = e^x - 8$

Key Properties of the Function:

1. Domain: The domain of $f(x)$ is all real numbers, $x \in \mathbb{R}$, since the exponential function $e^x$ is defined for all $x$.
2. Range: The range of $f(x)$ is $(-8, \infty)$, because $e^x \geq 1$ for all $x \geq 0$, and subtracting 8 lowers the range accordingly.
3. Intercept:
   • y-intercept: Plugging $x = 0$, we get $f(0) = e^0 - 8 = 1 - 8 = -7$.
   • x-intercept: Solving $f(x) = 0$: $e^x - 8 = 0 \implies e^x = 8 \implies x = \ln(8)$
4. Asymptote: The horizontal asymptote of $f(x)$ is $y = -8$ as $x \to -\infty$, since $e^x \to 0$ as $x \to -\infty$.
5. Behavior:
   • $f(x)$ increases exponentially as $x \to \infty$.
   • $f(x)$ decreases and approaches $-8$ as $x \to -\infty$.

Let me know if you'd like a graph of this function or further analysis!

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Related Questions:

1. How do you compute the derivative of $f(x)$?
2. What is the second derivative of $f(x)$ and its implications?
3. What are the critical points and intervals of increase/decrease for $f(x)$?
4. How does the graph of $e^x - 8$ compare to $e^x$?
5. What are the practical applications of functions like $f(x)$ in real-world contexts?

Tip: When solving equations with exponentials, always consider rewriting them in logarithmic form for simplicity!