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We are given the following functions:

- \( g(x) = \frac{1}{x} \)
- \( g(x)^2 = x^2 - 2x + 1 \)
- \( h(x) = x^2 - 2x - 3 \)

The question asks us to determine which function \( f(x) \) from the provided options satisfies the equation \( (f \circ g)(x) - h(x) = 0 \).

Let's break this down:

1. **Composition** \( (f \circ g)(x) \) means we substitute \( g(x) \) into \( f(x) \).
2. After substituting, subtract \( h(x) \).
3. Then, compare the result to zero to determine the function \( f(x) \).

I'll calculate this step by step. Let's start with composing \( f(g(x)) \) for each option and subtracting \( h(x) \), then simplify.

Let me work through this.Here are the simplified results for \( (f \circ g)(x) - h(x) \) for each candidate function \( f(x) \):

1. \( f(x) = x^2 - 4 \): \( -x^2 + 2x - 1 + \frac{1}{x^2} \)
2. \( f(x) = x - 3 \): \( \frac{x^2(2 - x) + 1}{x} \)
3. \( f(x) = -x^2 + 5 \): \( -x^2 + 2x + 8 - \frac{1}{x^2} \)
4. \( f(x) = x^2 - 3x - 2 \): \( -x^2 + 2x + 1 - \frac{3}{x} + \frac{1}{x^2} \)

We are looking for \( (f \circ g)(x) - h(x) = 0 \), but none of the options simplify exactly to zero. The one closest to zero without extra terms (except the rational fractions) is **Option 4**, \( f(x) = x^2 - 3x - 2 \).

Thus, the correct answer is likely:

\[
f(x) = x^2 - 3x - 2
\]

Would you like more details on this, or any further explanation? 

Here are 5 related questions to explore this further:
1. How do you solve composition of functions step by step?
2. Why are rational expressions involved in the results?
3. How can we check if a function is invertible for compositions?
4. How does substitution work in function compositions?
5. How do higher powers of \( x \) affect function behavior?

**Tip**: When dealing with function composition, always remember that order matters, and breaking it down step by step makes it clearer!

For which function f listed below is (f ∘ g)(x) = h(x)?

Learn how to solve a problem involving function composition, specifically determining f(x) where (f ∘ g)(x) = h(x). This problem focuses on algebra, function composition, and quadratic equations. Suitable for students in Grades 10-12.

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