HM

Notes on Decomposing Functions (Decomposition and f∘g)

Decomposition of a Function into Component Parts

  • Goal: Understand how to break a complex function into simpler building blocks by expressing it as a composition of two (or more) functions.
  • Key idea: Decomposition splits a big, potentially nested operation into outer and inner parts that can be studied separately.
  • Revisit of composition basics: If you have two functions, f and g, their composition is (f ∘ g)(x) = f(g(x)).
  • Practical takeaway: You can peel off outer layers of a function to define an outer function f and an inner function g so that the original function equals f(g(x)).
  • Analogy: Think of a function as a multi-layer cake; each layer is a building block you can detach and study or reassemble in a slightly different order.
  • Important note: There are many valid decompositions for a given big function; different choices reveal different building blocks and can simplify analysis or manipulation.
  • Exercise mindset: After seeing one decomposition, try to find alternative f and g pairs that yield the same overall function.
  • Real-world relevance: Decomposition helps in solving problems by isolating operations, optimizing computations, or preparing for further operations (like differentiation or integration) where a simpler inner structure is advantageous.
  • Chapter and future link: The idea of choosing different f and g is illustrated here; in this course, identity-based decompositions will be discussed later in Lesson 1.6, and the concept of the identity function will be explored there.
  • Ethical/philosophical/practical note: The practice emphasizes flexibility and verification; always check that your chosen f and g actually satisfy (f ∘ g)(x) = h(x) for the target function h.

Example 1: Decomposing a rational-radical function

  • Target big function (as a composition): h(x) = rac{1}{\,oxed{ \, oot 2 rom{(2x^2+1)} \,}} = rac{1}{ oot 2 rom{(2x^2+1)}} = rac{1}{\, ext{sqrt}(2x^2+1)}
    • Intuition: If you were to evaluate h(x) directly, the internal steps would be: square x, multiply by 2, add 1, take a square root, then take the reciprocal.
  • Step-by-step inner-to-outer view (the steps you’d apply in sequence):
    • y_1 = x^2
    • y2 = 2y1
    • y3 = y2 + 1
    • y4 = \, ext{sqrt}(y3)
    • h(x) = rac{1}{y_4}
  • Decomposition option A (two-component, outer f and inner g):
    • $$ f(x) = rac{1}{\, ext{sqrt}(x)} \