To determine the range of the function \( y = x - \frac{1}{x} \), we need to analyze its behavior over its domain.

### Step 1: Domain of the function
The function is defined for all real values of \( x \), except \( x = 0 \) (since division by zero is undefined). Thus, the domain of the function is:
\[ x \in (-\infty, 0) \cup (0, \infty) \]

### Step 2: Find critical points
We can find the critical points by differentiating \( y = x - \frac{1}{x} \) and setting the derivative equal to zero to find local minima and maxima.

#### Derivative:
\[ \frac{dy}{dx} = 1 + \frac{1}{x^2} \]

Set \( \frac{dy}{dx} = 0 \) to find critical points:
\[ 1 + \frac{1}{x^2} = 0 \]

This equation has no real solutions because \( \frac{1}{x^2} \) is always positive, and adding 1 will never equal 0. Therefore, there are no critical points where the derivative is zero.

### Step 3: Analyze the behavior at the boundaries
- As \( x \to 0^+ \), \( y = x - \frac{1}{x} \to -\infty \).
- As \( x \to 0^- \), \( y = x - \frac{1}{x} \to \infty \).
- As \( x \to \infty \), \( y = x - \frac{1}{x} \to \infty \).
- As \( x \to -\infty \), \( y = x - \frac{1}{x} \to -\infty \).

### Step 4: Minima of the function
We find the local minimum by evaluating \( y = x - \frac{1}{x} \) for specific values of \( x \). A good candidate to check is \( x = 1 \), because it often provides insight into the behavior of rational functions like this:

- When \( x = 1 \):
  \[
  y = 1 - \frac{1}{1} = 0
  \]
- When \( x = -1 \):
  \[
  y = -1 - \frac{1}{-1} = -2
  \]

Thus, the function has a minimum value of \( -2 \) at \( x = -1 \).

### Conclusion:
The range of \( y = x - \frac{1}{x} \) is:
\[
(-\infty, -2] \cup [0, \infty)
\]
This means the function takes on all values less than or equal to \( -2 \) and all values greater than or equal to 0.

Would you like more details on how to derive these results?

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Here are 5 related questions you might explore:

1. How would the range change if the function was \( y = x - \frac{2}{x} \)?
2. What is the behavior of the function as \( x \to \infty \) and \( x \to -\infty \)?
3. How can you find the range of more complex functions involving rational expressions?
4. What happens to the function if we add a constant, such as \( y = x - \frac{1}{x} + c \)?
5. How can you use the second derivative test to confirm the nature of the minimum at \( x = -1 \)?

**Tip:** When analyzing rational functions, always check for domain restrictions (like division by zero) and behavior at the boundaries!

Analyze the range of the rational function y = x - 1/x by finding its domain, derivative, and behavior at boundaries. This detailed solution explores critical points and the function's minimum value at x = -1. Suitable for students in Grades 10-12.

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