Math Problem Statement

For the following exercises, find functions f(x) and g(x) so the given functions can be expressed as h(x) = f(g(x)). There are multiple possible answers. If the answers require exponents, enter the answer with an exponent in f(x) but not in g(x). Given h(x) = 7 / (x + 2)^2, find f(x) and g(x).

Solution

We are tasked with decomposing the given function h(x)=7(x+2)2h(x) = \frac{7}{(x+2)^2} into two functions f(x)f(x) and g(x)g(x) such that h(x)=f(g(x))h(x) = f(g(x)).

One possible approach is to recognize that we can set g(x)g(x) as the inner function (involving xx) and f(x)f(x) as the outer function.

Let's break it down:

  1. Start by observing the expression h(x)=7(x+2)2h(x) = \frac{7}{(x+2)^2}.
  2. If we choose g(x)=x+2g(x) = x + 2, then the inner part g(x)g(x) is x+2x+2.
  3. Now, h(x)=7(g(x))2h(x) = \frac{7}{(g(x))^2}, so the outer function f(x)f(x) should be f(x)=7x2f(x) = \frac{7}{x^2}.

Thus, we have:

  • g(x)=x+2g(x) = x + 2
  • f(x)=7x2f(x) = \frac{7}{x^2}

These choices satisfy h(x)=f(g(x))h(x) = f(g(x)).

Would you like further details on this decomposition or any related questions? Here are some follow-up questions you might find useful:

  1. Can there be other possible decompositions for h(x)=f(g(x))h(x) = f(g(x))?
  2. How do we verify if our chosen f(x)f(x) and g(x)g(x) are correct?
  3. What other examples of function composition can we explore?
  4. How does the chain rule apply to derivatives of composite functions like f(g(x))f(g(x))?
  5. Can you create an example where the exponent appears only in g(x)g(x) and not in f(x)f(x)?

Tip: When decomposing functions, focus on identifying the inner and outer parts of the expression to make the process simpler.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra

Formulas

h(x) = f(g(x))
Exponent Rules

Theorems

Function Composition Theorem

Suitable Grade Level

Grades 10-12