https://r2.erweima.ai/mathbot/image/1727021113392vqnr9sb0.jpg

The function given in the image is a piecewise function:

\[
f(x) = \begin{cases} 
x^2 & \text{if } x < -2, \\
0 & \text{otherwise}.
\end{cases}
\]

We are asked to express this function in terms of the Heaviside function \( u(x) \), which is defined as:

\[
u(x) = \begin{cases} 
0 & \text{if } x < 0, \\
1 & \text{if } x \geq 0.
\end{cases}
\]

To rewrite \( f(x) \) using the Heaviside function, notice that \( f(x) = 0 \) when \( x \geq -2 \), and \( f(x) = x^2 \) when \( x < -2 \). The Heaviside function can help "turn off" the \( x^2 \) term when \( x \geq -2 \). This can be done using \( u(x+2) \), which is 1 when \( x \geq -2 \) and 0 when \( x < -2 \).

Thus, the compact form of the function using the Heaviside function is:

\[
f(x) = x^2 \cdot u(-x-2).
\]

Here, \( u(-x-2) \) turns on the \( x^2 \) term when \( x < -2 \), as desired.

Would you like more details or have further questions?

### Related Questions:
1. How do we define the Heaviside function in different contexts?
2. What is the significance of the Heaviside function in signal processing?
3. Can the Dirac delta function be related to the Heaviside function?
4. How do we compute derivatives of piecewise functions involving Heaviside functions?
5. How would this function look if we shifted the domain limits (e.g., \( x < -3 \) instead of \( x < -2 \))?

**Tip:** When working with piecewise functions, the Heaviside function is a powerful tool for converting them into a more compact, single expression.

La función f(x) = {x^2 si x < -2, 0 de otro modo} se puede expresar de forma compacta usando la función de Heaviside u(x).

Learn how to express the piecewise function f(x) = {x^2 if x < -2, 0 otherwise} in a compact form using the Heaviside function u(x). This solution covers the application of the Heaviside function and its properties.

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