plot the function given by f(x) = 3x-3+(-x^2 -3x+1)h(x) +(-2+x^2)h(x-2), where h(x) is the Heaviside step function

Here is the plot of the function \( f(x) = 3x - 3 + (-x^2 - 3x + 1)h(x) + (-2 + x^2)h(x - 2) \), where \( h(x) \) is the Heaviside step function.

The key changes occur at \( x = 0 \) and \( x = 2 \), where the step function activates, altering the behavior of the function.

Would you like any further details or explanations?

Here are some related questions:
1. How does the Heaviside step function influence piecewise functions?
2. What happens if the step is applied at different points (e.g., \( h(x - 1) \))?
3. Can this function be represented using piecewise notation?
4. What is the significance of the terms in this function for each interval?
5. How does this function behave as \( x \to \infty \)?

**Tip:** The Heaviside step function is often used to model sudden changes or "switching" behavior in physical systems and control theory.

Plot the function given by f(x) = 3x - 3 + (-x^2 - 3x + 1)h(x) + (-2 + x^2)h(x - 2), where h(x) is the Heaviside step function.

Explore how to plot the function f(x) = 3x - 3 + (-x^2 - 3x + 1)h(x) + (-2 + x^2)h(x - 2), where the Heaviside step function causes key shifts in behavior. This problem involves step functions, piecewise functions, and quadratic equations. Suitable for students in Grades 10-12.

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