Let's consider the properties of the given exponential function:

Properties:

1. Domain: $\mathbb{R}$ (all real numbers).
2. Range: $y > 0$ (the function's values are always positive).
3. Y-intercept: 3 (the function crosses the y-axis at $y = 3$).
4. Horizontal asymptote: $y = 0$ (the function approaches zero but never touches or crosses it as $x \to \infty$).
5. Decreasing function: The function is decreasing, meaning the function's value gets smaller as $x$ increases.

General Form:

The general form of an exponential function is:

$y = ab^x$

For the function to be decreasing, $0 < b < 1$. The y-intercept is 3, meaning when $x = 0$, $y = 3$, so $a = 3$. Therefore, the function has the form:

$y = 3b^x$

where $0 < b < 1$.

Graph Sketch:

• The graph will pass through the point $(0, 3)$.
• As $x \to \infty$, the function will approach $y = 0$ but never touch the x-axis (the horizontal asymptote).
• As $x \to -\infty$, the function will increase rapidly.
• The graph is decreasing since $0 < b < 1$.

I can generate a sketch for this.

Part (b):

Is it possible to sketch another exponential function with the same properties?

No, it is not possible to sketch another function with the same properties. The key constraints, such as the domain $\mathbb{R}$, range $y > 0$, the horizontal asymptote at $y = 0$, and the decreasing behavior, are all uniquely described by the function $y = 3b^x$ with $0 < b < 1$. Changing any of these would violate one or more of the given conditions.

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Would you like me to generate a visual sketch of this graph? Also, do you want more details on any part?

Here are five related questions to expand your understanding:

1. How does changing the base $b$ affect the steepness of the graph?
2. Can an exponential function have a horizontal asymptote at any value other than zero?
3. What is the behavior of the function as $x \to -\infty$?
4. How does the graph of a decreasing exponential function compare to that of an increasing one?
5. How would you transform this function to move the asymptote to a different y-value?

Tip: To identify the behavior of an exponential function, pay close attention to the base $b$ and the asymptote—it tells you a lot about the growth or decay pattern.