Math Problem Statement

Express the given function h(x) as a composition of two functions f and g so that h(x) = (f ∘ g)(x), where one of the functions is 7x - 2.

Solution

The problem asks to express the given function h(x)=(7x2)8h(x) = (7x - 2)^8 as a composition of two functions ff and gg, where h(x)=(fg)(x)=f(g(x))h(x) = (f \circ g)(x) = f(g(x)), and one of the functions is 7x27x - 2.

To solve this:

  1. We are told that one of the functions should be g(x)=7x2g(x) = 7x - 2. Thus, g(x)=7x2g(x) = 7x - 2.
  2. Next, we need to define f(x)f(x) such that f(g(x))=h(x)f(g(x)) = h(x). Notice that h(x)h(x) is (7x2)8(7x - 2)^8, and g(x)=7x2g(x) = 7x - 2, so h(x)h(x) is f(g(x))=f(7x2)f(g(x)) = f(7x - 2).
  3. Therefore, f(x)f(x) should take an input and raise it to the 8th power. This means f(x)=x8f(x) = x^8.

Thus, the composition is:

  • g(x)=7x2g(x) = 7x - 2
  • f(x)=x8f(x) = x^8

So, h(x)=(fg)(x)=f(g(x))=(7x2)8h(x) = (f \circ g)(x) = f(g(x)) = (7x - 2)^8.

Would you like further details or clarification on any part of this solution? Here are some related questions:

  1. How do you determine the inner and outer functions in a composition?
  2. Can you generalize this method for other functions that are powers of linear expressions?
  3. What are some common applications of function composition in real-world scenarios?
  4. How do you differentiate a composition of functions using the chain rule?
  5. What happens if the inner function is more complex than a linear expression?

Tip: When composing functions, always try to identify which part of the original function can be isolated and treated as the "inner" function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Exponents
Algebra

Formulas

h(x) = (f ∘ g)(x) = f(g(x))
g(x) = 7x - 2
f(x) = x^8

Theorems

Composition of Functions

Suitable Grade Level

Grades 9-12