Which of the following formulas could represent a function with a domain of all Real numbers?


y=(1)/(x),  y=|x|,  y=\sqrt(x),   x=5

Let's analyze each of the given formulas to determine if they could represent a function with a domain of all real numbers:

1. **\( y = \frac{1}{x} \)**:
   - This is a rational function, and it is undefined when \( x = 0 \) because division by zero is not possible.
   - **Domain**: \( \mathbb{R} \setminus \{0\} \) (all real numbers except 0).

2. **\( y = |x| \)**:
   - This is the absolute value function, which is defined for all real numbers \( x \).
   - **Domain**: \( \mathbb{R} \) (all real numbers).

3. **\( y = \sqrt{x} \)**:
   - This is a square root function, and it is only defined for non-negative values of \( x \) (i.e., \( x \geq 0 \)).
   - **Domain**: \( [0, \infty) \) (all non-negative real numbers).

4. **\( x = 5 \)**:
   - This represents a vertical line, which is not a function because it does not satisfy the definition of a function where each \( x \) value must map to exactly one \( y \) value. Instead, it represents a specific value of \( x \) and does not involve \( y \) as a dependent variable.

### Conclusion
The only formula that represents a function with a domain of all real numbers is **\( y = |x| \)**.

Would you like more details on this, or do you have any questions?

Here are 5 related questions to consider:
1. What is the range of the absolute value function \( y = |x| \)?
2. How do you graph the function \( y = |x| \) and what are its key features?
3. How can you determine if a given equation represents a function?
4. What transformations can be applied to \( y = |x| \) to change its graph?
5. What are some examples of other functions with domains of all real numbers?

**Tip:** When analyzing a function, remember that the domain consists of all possible input values for which the function is defined. Always check for restrictions like division by zero or square roots of negative numbers.

Explore different mathematical formulas and their domains, including rational functions, absolute value functions, and square root functions. Learn how to determine which formulas represent functions with domains of all real numbers.

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