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The question is asking to find the correct pair of functions \( f(x) \) and \( g(x) \) such that \( h(x) = (f \circ g)(x) = f(g(x)) \), and given that \( h(x) = -\sqrt{2x^2 - 1} + 3 \).

We need to find the functions \( f(x) \) and \( g(x) \) that, when composed as \( f(g(x)) \), give the function \( h(x) \).

Start by analyzing the structure of \( h(x) \):
\[
h(x) = -\sqrt{2x^2 - 1} + 3
\]
This resembles the form of a composition where:
- Inside the square root is \( 2x^2 - 1 \), which suggests \( g(x) = 2x^2 - 1 \).
- The outer function is \( -\sqrt{x} + 3 \), which suggests \( f(x) = -\sqrt{x} + 3 \).

Thus, we have the functions:
\[
g(x) = 2x^2 - 1
\]
\[
f(x) = -\sqrt{x} + 3
\]

Looking at the provided options, the correct choice is the third one:
\[
f(x) = -\sqrt{x} + 3, \quad g(x) = 2x^2 - 1
\]

Would you like further details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the definition of function composition in mathematics?
2. How do you determine the functions \( f \) and \( g \) in a composite function?
3. What is the domain of \( h(x) \) in this example, given the square root function involved?
4. How would the graph of \( h(x) \) compare to the graphs of \( f(x) \) and \( g(x) \)?
5. Can function composition always be reversed to find \( f \) and \( g \) from \( h(x) \)?

**Tip**: When working with composite functions, always check the domain of the inner function, as it can affect the domain of the overall composition.

The function h(x) = -√(2x^2 - 1) + 3 can be written as (f ∘ g)(x) with f(x) and g(x). Select the correct pair of functions f(x) and g(x) from the given options.

Learn how to find the correct pair of functions f(x) and g(x) for the composition h(x) = -√(2x^2 - 1) + 3. This problem involves function composition, quadratic functions, and square roots. Suitable for high school students in Grades 10-12.

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