y(0) y(x)=x+1/x what happens to the function as x increases? what happens to the function when x decreases

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.

Explore the behavior of the function y(x) = x + 1/x as x increases and decreases. Learn about its increasing and decreasing trends, critical points, and graphical behavior. Suitable for advanced high school students and above.

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