Indicate whether the two functions are equal. If the two functions are not equal, then give an element of the domain on which the two functions have different values. ||
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![Indicate whether the two functions are equal. If the two functions are not equal, then give an element of the domain on which the two functions have different values.
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![**Transcription for Educational Website**
---
**Functions Defined on Integers**
1. **Function \( f \)**:
\[
f : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}, \text{ where } f(x, y) = |x + y|
\]
- **Description**: The function \( f \) takes two integer inputs, \( x \) and \( y \), and maps them to the integer which is the absolute value of their sum.
2. **Function \( g \)**:
\[
g : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}, \text{ where } g(x, y) = |x| + |y|
\]
- **Description**: The function \( g \) is defined similarly to \( f \), but instead of summing \( x \) and \( y \) before taking the absolute value, it computes the absolute values of \( x \) and \( y \) individually and then sums these two results.
**Explanation of Notation**:
- \( \mathbb{Z} \) denotes the set of all integers.
- The functions \( f \) and \( g \) map pairs of integers to a single integer.
- \( | \cdot | \) denotes the absolute value function, which returns the non-negative magnitude of a number.
**Key Concepts**:
- **Absolute Value**: The absolute value of a number is its distance from zero on the number line, regardless of direction.
- **Sum Function \( f \)**: Focuses on the combination of two numbers and the magnitude of their sum.
- **Component-wise Absolute Sum Function \( g \)**: Focuses on magnitude separately before combining.
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These functions illustrate different ways of combining two integer inputs to produce another integer, highlighting the use of absolute value in various contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27f445ea-1d9a-48bd-a007-85d03e874c3c%2Fbc5c5b89-ee98-47d0-b195-0bad68e386c9%2Fl47qzzj_processed.png&w=3840&q=75)
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