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!