To determine the range of the function \( f(x) = \cos^2 x - 2\cos x \), let's proceed step by step.

### Step 1: Express \( f(x) \) in terms of \( y = \cos x \)
Let \( y = \cos x \). The function then becomes:
\[
f(x) = y^2 - 2y
\]

### Step 2: Consider the range of \( y \)
Since \( y = \cos x \), the possible values of \( y \) are between \(-1\) and \( 1 \), i.e., \( -1 \leq y \leq 1 \).

### Step 3: Analyze the quadratic function
The function \( f(y) = y^2 - 2y \) is a quadratic function. To determine its range, we first find the vertex, since a quadratic function opens upwards (coefficient of \( y^2 \) is positive) and its minimum value occurs at the vertex.

The vertex of the quadratic function \( f(y) = y^2 - 2y \) occurs at:
\[
y_{\text{vertex}} = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1
\]
So, the minimum value of \( f(y) \) is at \( y = 1 \):
\[
f(1) = (1)^2 - 2(1) = 1 - 2 = -1
\]

### Step 4: Calculate \( f(y) \) at the boundaries
Evaluate \( f(y) \) at the boundaries \( y = -1 \) and \( y = 1 \):

- For \( y = -1 \):
\[
f(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3
\]

- For \( y = 1 \):
\[
f(1) = 1 - 2 = -1
\]

### Step 5: Determine the range
The function \( f(y) = y^2 - 2y \) takes on values between \(-1\) and \( 3 \) as \( y \) varies from \(-1\) to \( 1 \).

Therefore, the range of \( f(x) \) is:
\[
\boxed{[-1, 3]}
\]

Would you like any further details or have any questions?

Here are five related questions to explore further:

1. What is the domain of \( f(x) = \cos^2 x - 2\cos x \)?
2. How would you find the range of \( f(x) = \sin^2 x + 3\sin x \)?
3. Can you determine the maximum value of the function \( f(x) = \cos^2 x - 2\cos x \)?
4. What transformations would apply to \( f(x) = \cos x \) to obtain \( f(x) = \cos^2 x - 2\cos x \)?
5. How does the range of \( f(x) = \cos^2 x - 2\cos x \) compare with that of \( f(x) = \cos^2 x \)?

**Tip:** When analyzing the range of a function, especially when involving trigonometric identities or quadratic expressions, consider substitution to simplify and reduce the problem to a familiar form like a quadratic equation.

Discover how to find the range of the function f(x) = cos^2 x - 2 cos x using trigonometric identities and quadratic function analysis. This problem involves understanding cosine functions, quadratic equations, and their transformations. Suitable for students in Grades 10-12.

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