The function $F(x) = 3^x - 1$ is an exponential function with a vertical shift. Let’s break it down:

1. Base function: $3^x$, which grows exponentially as $x$ increases.
2. Vertical shift: The term $-1$ shifts the graph downward by 1 unit.

Key Properties:

1. Domain: The function is defined for all real numbers, so $\text{Domain} = (-\infty, \infty)$.
2. Range: Since $3^x > 0$ for all $x$, the range of $F(x) = 3^x - 1$ is $(-1, \infty)$.
3. Intercepts:
   • Y-intercept: Set $x = 0$, so $F(0) = 3^0 - 1 = 1 - 1 = 0$.
   • No X-intercept other than $x = 0$, as $3^x$ never equals 1 for negative values of $x$.
4. Behavior as $x \to -\infty$: $3^x \to 0$, so $F(x) \to -1$.
5. Asymptote: The horizontal asymptote is $y = -1$.

Would you like a graph, more details, or specific applications? Let me know!

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Here are 5 related questions to explore:

1. What is the derivative of $F(x) = 3^x - 1$?
2. How does $F(x) = 3^x - 1$ compare to $G(x) = 2^x - 1$?
3. What is the inverse function of $F(x) = 3^x - 1$?
4. How do you solve $3^x - 1 = 5$?
5. How does a vertical shift affect an exponential function graphically?

Tip: For exponential functions, the base (e.g., 3 in $3^x$) determines the growth rate!