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.