Math Problem Statement

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.

Solution

The question is asking to find the correct pair of functions f(x)f(x) and g(x)g(x) such that h(x)=(fg)(x)=f(g(x))h(x) = (f \circ g)(x) = f(g(x)), and given that h(x)=2x21+3h(x) = -\sqrt{2x^2 - 1} + 3.

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

Start by analyzing the structure of h(x)h(x): h(x)=2x21+3h(x) = -\sqrt{2x^2 - 1} + 3 This resembles the form of a composition where:

  • Inside the square root is 2x212x^2 - 1, which suggests g(x)=2x21g(x) = 2x^2 - 1.
  • The outer function is x+3-\sqrt{x} + 3, which suggests f(x)=x+3f(x) = -\sqrt{x} + 3.

Thus, we have the functions: g(x)=2x21g(x) = 2x^2 - 1 f(x)=x+3f(x) = -\sqrt{x} + 3

Looking at the provided options, the correct choice is the third one: f(x)=x+3,g(x)=2x21f(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 ff and gg in a composite function?
  3. What is the domain of h(x)h(x) in this example, given the square root function involved?
  4. How would the graph of h(x)h(x) compare to the graphs of f(x)f(x) and g(x)g(x)?
  5. Can function composition always be reversed to find ff and gg from h(x)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.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra
Square Roots
Quadratic Functions

Formulas

h(x) = f(g(x))
g(x) = 2x^2 - 1
f(x) = -√x + 3

Theorems

Function Composition

Suitable Grade Level

Grades 10-12