The function $f(x) = 76 (1.032)^x$ represents exponential growth. Here's a detailed analysis of the function:

1. Initial Value: The initial value of the function, when $x = 0$, is $f(0) = 76$. This is the starting value of the function.

2. Growth Factor: The base of the exponential term, $1.032$, is the growth factor. It indicates that the quantity increases by $3.2\%$ per unit of $x$.

3. Rate of Growth: The growth rate is derived from the growth factor: $\text{Growth Rate} = (1.032 - 1) \times 100\% = 3.2\%.$

4. Behavior: As $x \to \infty$, $f(x) \to \infty$, which means the function grows indefinitely. As $x \to -\infty$, $f(x) \to 0^+$, approaching zero but never reaching it.

5. Graph Characteristics:

   • Shape: The graph is upward-sloping and becomes steeper as $x$ increases, illustrating rapid growth over time.
   • Intercept: The y-intercept is at $(0, 76)$.

Example Calculations:

• At $x = 1$: $f(1) = 76 \cdot (1.032)^1 = 78.432$.
• At $x = 5$: $f(5) = 76 \cdot (1.032)^5 \approx 89.045$.
• At $x = -2$: $f(-2) = 76 \cdot (1.032)^{-2} \approx 71.343$.

Would you like further examples or a graph of this function?

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Related Questions:

1. What is the doubling time of this exponential function?
2. How does the function behave when the growth factor changes to $1.05$?
3. What is the derivative of this function, and what does it tell us?
4. Can you compare this growth to linear growth with the same initial value and rate?
5. How would the function look with a decay factor instead, e.g., $0.97^x$?

Tip:

To find when the value doubles, solve $76 \cdot 1.032^x = 152$.