https://r2.erweima.ai/mathbot/image/1727021072837u61fugql.jpg

The problem asks for a compact expression of the given function \( f(x) \) using the Heaviside step function \( u(x) \).

The function \( f(x) \) is defined as:

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

The Heaviside step function, \( u(x) \), is defined as:

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

To represent \( f(x) \) using the Heaviside function, we need to turn off the \( x^2 \) term for \( x \geq -2 \) and leave it active only when \( x < -2 \). This can be done using the shift of the Heaviside function.

The correct form will be:

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

Here, \( u(-2 - x) \) ensures that \( x^2 \) is active when \( x < -2 \) and 0 otherwise.

Would you like to ask any further questions or get more details?

### Related questions:
1. What is the definition of the Heaviside step function \( u(x) \) in more detail?
2. How can we use the Heaviside function for a more complex piecewise function?
3. Can we represent any function with discontinuities using the Heaviside function?
4. How would the graph of the function \( f(x) = x^2 \cdot u(-2 - x) \) look?
5. What are the applications of the Heaviside step function in engineering and physics?

### Tip:
When working with the Heaviside function, a common technique is to shift the argument to control where the step occurs.

La función f(x) = { x^2 si x < -2; 0 de otra manera } se puede expresar de forma compacta usando la función de Heaviside u(x). Elige la opción correcta.

Learn how to express the piecewise function f(x) = { x^2 if x < -2; 0 otherwise } in a compact form using the Heaviside step function u(x). This tutorial covers key algebraic concepts and shows how to use the Heaviside function to handle discontinuities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation