Given the function $y(x) = x + \frac{1}{x}$, let's analyze what happens as $x$ increases and decreases.

1. As $x$ increases:

• When $x$ increases, the term $x$ grows larger.
• The term $\frac{1}{x}$ decreases since it is the reciprocal of $x$, which gets smaller as $x$ gets larger.
• Therefore, the function $y(x)$ will continue to increase because the increasing $x$ term dominates over the decreasing $\frac{1}{x}$ term.

2. As $x$ decreases:

• When $x$ decreases but remains positive (i.e., $x > 0$), the term $x$ becomes smaller.

• The term $\frac{1}{x}$ becomes larger as $x$ gets closer to zero.

• As $x$ approaches zero from the positive side, $\frac{1}{x}$ tends to infinity, and thus $y(x)$ will increase rapidly.

• If $x$ is negative, both $x$ and $\frac{1}{x}$ are negative. However, $x$ will dominate and tend to decrease as $x$ becomes more negative, making $y(x)$ more negative as well.

Summary:

• As $x$ increases, $y(x)$ increases.
• As $x$ decreases (in the positive domain), $y(x)$ increases rapidly as $x$ approaches zero.
• For large negative $x$, $y(x)$ decreases.

Would you like more details on this function, such as its critical points or graphical behavior?

Here are five related questions to explore further:

1. What is the derivative of $y(x) = x + \frac{1}{x}$, and what does it tell us about the function's behavior?
2. What are the critical points of the function $y(x) = x + \frac{1}{x}$?
3. How does the function behave as $x$ approaches negative infinity?
4. What is the minimum value of the function $y(x) = x + \frac{1}{x}$ for $x > 0$?
5. How does the function behave around $x = -1$?

Tip: When analyzing functions, it's useful to consider both the first and second derivatives to understand the function's increasing/decreasing behavior and concavity.