The problem is to analyze the exponential function $f(x) = 3 \cdot 2^x$. The image shows the graph coordinate system, and the function is plotted.

1. Problem Description

The problem is to analyze the exponential function $f(x) = 3 \cdot 2^x$. The image shows the graph coordinate system, and the function is plotted.

2. Solution Steps

To understand the function, we can evaluate it for a few values of $x$:

If $x = 0$, then $f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3$.

If $x = 1$, then $f(1) = 3 \cdot 2^1 = 3 \cdot 2 = 6$.

If $x = 2$, then $f(2) = 3 \cdot 2^2 = 3 \cdot 4 = 12$.

If $x = -1$, then $f(-1) = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = \frac{3}{2} = 1.5$.

If $x = -2$, then $f(-2) = 3 \cdot 2^{-2} = 3 \cdot \frac{1}{4} = \frac{3}{4} = 0.75$.

The graph will pass through the points $(0, 3)$, $(1, 6)$, $(2, 12)$, $(-1, 1.5)$, and $(-2, 0.75)$. The graph will increase rapidly as $x$ increases, and approach $0$ as $x$ goes to negative infinity.

3. Final Answer

The function is $f(x) = 3 \cdot 2^x$. We can analyze the behavior of this exponential function by plugging in some values of x.