Let's analyze the function $f(x) = -2 \cdot 2^{\frac{1}{3} x} + 3$ and fill in each characteristic:

1. Domain: The function $f(x) = -2 \cdot 2^{\frac{1}{3} x} + 3$ is an exponential function, which is defined for all real values of $x$.

   • Domain: $(-\infty, \infty)$

2. Range: Since the function is of the form $a \cdot b^{cx} + d$, where $a = -2$ (negative value) and $d = 3$, it has a horizontal asymptote at $y = 3$ and will approach this value from below as $x \to \infty$. This makes the range $(-\infty, 3)$.

   • Range: $(-\infty, 3)$

3. y-intercept: To find the $y$-intercept, set $x = 0$: $f(0) = -2 \cdot 2^{\frac{1}{3} \cdot 0} + 3 = -2 \cdot 1 + 3 = 1$

   • y-intercept: $(0, 1)$

4. Asymptote: The function has a horizontal asymptote as $y$ approaches the constant term, $y = 3$, as $x \to \infty$.

   • Asymptote: $y = 3$

5. Increasing/Decreasing Intervals: Since $a = -2$ is negative, the function is decreasing for all $x$.

   • Increasing Interval: $\emptyset$ (null set)
   • Decreasing Interval: $(-\infty, \infty)$

6. End Behavior:

   • As $x \to -\infty$, $f(x) \to -\infty$ because the exponential term $2^{\frac{1}{3}x}$ approaches zero, making $f(x)$ go to $-\infty$.

   • As $x \to \infty$, $f(x) \to 3$ (approaching the horizontal asymptote from below).

   • End Behavior:

     • As $x \to -\infty$, $f(x) \to -\infty$
     • As $x \to \infty$, $f(x) \to 3$

Would you like further explanation on any of these characteristics?

Here are 5 additional questions that might interest you:

1. How do you find the range of an exponential function?
2. Why does a negative coefficient affect the direction of an exponential function's growth or decay?
3. What other types of functions have horizontal asymptotes?
4. How does the base of the exponent affect the shape of the graph?
5. Can an exponential function ever cross its horizontal asymptote?

Tip: When graphing exponential functions, always identify the horizontal asymptote first to understand the behavior of the function.