Find functions f and g such that h=g ∘f . h(x)=(2 x^3+x^2+1)^5

Name: Problem 35, Chapter 2: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-06-30T05:40:19-08:00
Duration: 1 min 31 s
Channel: Manisha Sarker
Description: Find functions f and g such that h=g ∘f . h(x)=(2 x^3+x^2+1)^5

step 1 In this problem, we have been given a function H of X, and it is said that this is equal to 2xq

Find functions f and g such that h=g ∘f . h(x)=(2...

Key Concepts

Function Composition

Function composition is the process of applying one function to the output of another, forming a new function. It is denoted as (g ? f)(x) = g(f(x)). This concept is key for understanding how to break down or build up complex functions by organizing operations sequentially, thereby simplifying analysis and computation.

Function Decomposition

Function decomposition involves breaking a complex function into simpler, constituent functions whose composition yields the original function. This technique is useful for analyzing and simplifying complex functions by identifying natural sub-functions or layers within the overall structure.

Identifying Inner and Outer Functions

Identifying inner and outer functions is crucial in function composition. The inner function is applied first and its result is then used as the input for the outer function. Recognizing this hierarchical structure aids in solving problems where a composite function is given and one must determine the constituent functions.

Polynomial Functions

Polynomial functions are expressions composed of variables and coefficients involving operations of addition, subtraction, and non-negative integer exponents. In function decomposition, recognizing polynomial components can facilitate the separation of parts of a composite function, especially when they are combined with other operations such as exponentiation.

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