Let's analyze and solve the given function:

The function:

$f(x) = -0.2x^3 + 4$

1. Graphing using Transformations:

• The basic function is $f(x) = x^3$, which is a cubic function.
• The coefficient $-0.2$ indicates a vertical compression (by a factor of 0.2) and reflection over the x-axis.
• The "+4" means a vertical shift upwards by 4 units.

So, the transformed graph will be a compressed, reflected cubic graph, shifted up by 4 units.

2. Domain:

The domain of any cubic function is all real numbers, since cubic functions extend infinitely in both directions horizontally. $\text{Domain}: (-\infty, \infty)$

3. Range:

Similarly, the range of a cubic function is also all real numbers, as the function extends infinitely both up and down. $\text{Range}: (-\infty, \infty)$

4. Horizontal Asymptote:

A cubic function generally does not have a horizontal asymptote because it extends infinitely in both positive and negative directions. However, since the highest degree term dominates the behavior of the function at extreme values of $x$, there is no horizontal asymptote for this function. $\text{Horizontal Asymptote}: \text{None}$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$
• Horizontal Asymptote: None

Would you like a step-by-step explanation of how to graph this function? Or perhaps details on transformations?

Related Questions:

1. How would the graph change if the function were $f(x) = 0.2x^3 + 4$?
2. What is the effect of changing the cubic term's coefficient to -1 on the graph?
3. Can you explain the role of vertical shifts in transforming functions?
4. What would be the domain and range of $f(x) = x^3 - 4x + 1$?
5. How do horizontal shifts affect the cubic graph $f(x) = x^3$?

Tip:

When graphing functions, always look at the leading term (highest degree) to determine long-term behavior and at transformations like vertical shifts or stretches for local modifications.