For what domain is the range of y=-x and y=-|x|

To determine the domain for which the range of \( y = -x \) and \( y = -|x| \) are the same, let's analyze the functions:

1. **For \( y = -x \):**
   - This is a linear function with a negative slope.
   - The domain of \( y = -x \) is all real numbers \( x \in (-\infty, \infty) \).
   - The range is also all real numbers \( y \in (-\infty, \infty) \).

2. **For \( y = -|x| \):**
   - This is the negative of the absolute value function.
   - The domain of \( y = -|x| \) is also all real numbers \( x \in (-\infty, \infty) \).
   - The range, however, is non-positive values, \( y \leq 0 \), because the absolute value is always non-negative, and taking the negative of that gives us non-positive values.

To find where the ranges of \( y = -x \) and \( y = -|x| \) are the same, we need \( -x \) to be non-positive. This happens when \( x \geq 0 \).

**Conclusion:**  
The domain for which the range of \( y = -x \) and \( y = -|x| \) are the same is \( x \geq 0 \). 

Would you like further details or have any questions?

Here are 5 related questions you might find useful:

1. What is the range of the function \( y = |x| \)?
2. How do you graph the function \( y = -|x| \)?
3. What happens to the function \( y = -x \) when you restrict the domain to \( x \geq 0 \)?
4. How do you find the intersection points of two functions graphically?
5. Can the function \( y = -|x| \) be expressed as a piecewise function?

**Tip:** Remember, when dealing with absolute value functions, always consider the sign of the input variable to properly evaluate the function!

Explore the domain and range analysis of functions y = -x and y = -|x| to understand where their ranges coincide. This problem covers functions, absolute value, and domain and range concepts in mathematics. Suitable for Grades 10-12.

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