The problem asks you to identify true statements regarding the function $f(x) = 2(3)^x$. Let’s analyze it:

Step-by-Step Analysis:

1. Exponential Growth: The function $f(x) = 2(3)^x$ is an exponential growth function because the base $3 > 1$. This means $f(x)$ increases as $x$ increases.

2. Constant Percent Rate of Change: In exponential functions of the form $f(x) = a(b)^x$, the constant percent rate of change is determined by the base $b$. Here, $b = 3$, so the graph increases by a constant percent rate of change, not decreases.

3. Behavior as $x \to \infty$: As $x \to \infty$, $f(x) \to \infty$, so $y$ increases without bound.

4. Behavior as $x \to -\infty$: As $x \to -\infty$, $(3)^x \to 0$, so $f(x) = 2 \cdot 0 = 2$. The value of $y$ approaches $2$.

Correct Answers:

• The graph is increasing by a constant percent rate of change of $3$.
• As $x$ increases, $y$ increases.
• As $x$ decreases, $y$ approaches $2$.

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5 Related Questions:

1. What is the constant percent rate of change in an exponential function with base $b = 4$?
2. How does the graph of $f(x) = 2(0.5)^x$ behave compared to $f(x) = 2(3)^x$?
3. What happens to the function $f(x) = 5(3)^x + 2$ as $x \to -\infty$?
4. Why does $y$ approach $2$ as $x \to -\infty$ in this problem?
5. How would changing the coefficient $2$ in $f(x) = 2(3)^x$ affect the graph?

Math Tip:

Always identify the base of an exponential function to determine whether it represents growth ($b > 1$) or decay ($0 < b < 1$). This helps predict the behavior of the graph.