To find the range of the function \( f(x) = \left(\frac{3}{x} - 1\right)^2 \), let's break it down step by step:

### Step 1: Analyze the structure
The function is composed of a rational term, \( \frac{3}{x} \), and a shift by \( -1 \), followed by squaring. Since squaring any real number yields a non-negative result, we know that:

\[
f(x) = \left(\frac{3}{x} - 1\right)^2 \geq 0
\]

### Step 2: Identify any restrictions on the function
The only restriction in this function is that \( x \neq 0 \), since division by zero is undefined. However, for any other real value of \( x \), the function is defined.

### Step 3: Range analysis
Let’s focus on \( \frac{3}{x} - 1 \). As \( x \) varies over all real numbers (except 0), \( \frac{3}{x} \) can take on all real values except zero. Subtracting 1 from these values still covers all real numbers, both negative and positive. However, when this expression is squared, the result is always non-negative.

Thus, the minimum value of \( f(x) \) occurs when \( \frac{3}{x} - 1 = 0 \), which gives \( f(x) = 0 \). This happens when \( \frac{3}{x} = 1 \), or \( x = 3 \).

### Step 4: Conclusion
The function has no upper bound since squaring larger values of \( \frac{3}{x} - 1 \) will yield increasingly large results. Therefore, the range of the function is:

\[
\boxed{[0, \infty)}
\]

Do you want further details or have any questions?

### Related Questions:
1. What is the domain of the function \( f(x) = \left(\frac{3}{x} - 1\right)^2 \)?
2. How would the range change if the function were \( \left(\frac{3}{x} + 1\right)^2 \)?
3. Can you find the range of \( f(x) = \left(\frac{5}{x} - 2\right)^2 \)?
4. How does the function behave as \( x \) approaches 0 from the positive and negative sides?
5. How do transformations (shifts and stretches) affect the range of squared functions?

### Tip:
Squaring a real number always results in a non-negative value, which is key to determining the range of any squared expression.

Explore how to determine the range of the function f(x) = (3/x - 1)^2 through a detailed breakdown of rational expressions and non-negative values from squaring real numbers. This algebraic concept is perfect for students in Grades 9-12.

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