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The given function is \( f(x) = x + \frac{1}{x} \). We are tasked with finding the set of values for which \( f(x) \) increases. To do this, we need to:

1. Differentiate \( f(x) \) to find its derivative, \( f'(x) \).
2. Determine where \( f'(x) > 0 \), which will give us the interval(s) where \( f(x) \) is increasing.

### Step 1: Find the derivative of \( f(x) \)
The function is:
\[
f(x) = x + \frac{1}{x}
\]
Differentiating this with respect to \( x \):
\[
f'(x) = \frac{d}{dx} \left( x + \frac{1}{x} \right) = 1 - \frac{1}{x^2}
\]

### Step 2: Solve \( f'(x) > 0 \)
Now we solve for when \( f'(x) > 0 \):
\[
1 - \frac{1}{x^2} > 0
\]
Rearranging the inequality:
\[
\frac{1}{x^2} < 1
\]
Taking the reciprocal and square-rooting both sides:
\[
|x| > 1
\]
This means \( x > 1 \) or \( x < -1 \).

### Conclusion
The function \( f(x) = x + \frac{1}{x} \) is increasing for \( x \in (-\infty, -1) \cup (1, \infty) \).

Would you like further details or have any questions?

### Related Questions:
1. How do we find where a function decreases?
2. What happens at the points \( x = 1 \) and \( x = -1 \)?
3. How do you determine whether a critical point is a local minimum or maximum?
4. What is the role of concavity in determining where a function is increasing or decreasing?
5. How can we use the second derivative to understand the behavior of \( f(x) \)?

**Tip:** Always check the sign of the derivative in different intervals to determine where the function increases or decreases.

If f(x) = x + 1/x, then the set of values for which f increases is

Learn how to determine the intervals where the function f(x) = x + 1/x is increasing. This problem involves differentiation and solving inequalities, making it a great example for high school students in Grades 10-12.

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