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1. Basic Concepts & Pre-requisites

A function f: A → B is a special relation where every element in domain A has exactly one unique image in codomain B.

Crucial Condition for Inverse Existence: A function must be a One-to-One Onto function (bijective) for its inverse to exist. If it is many-to-one or into, the inverse relation will fail the function test.

2. Composite Functions

When two functions combine such that the output of the first function becomes the input for the second, it forms a composite function.

(g o f)(x) = g(f(x))
(f o g)(x) = f(g(x))

Workflow to Evaluate (g o f)(x):

  1. Write down the outer function layout: g( ).
  2. Substitute the entire expression of the inner function f(x) inside the brackets of g.
  3. Expand and simplify the algebraic terms completely.

Note: Operational direction moves from right to left! f operates first, then g.

3. Inverse Functions

An inverse function reverses the operational direction, mapping the codomain elements back onto their original domain values.

If f(x) = y, then f⁻¹(y) = x

Standard Algebraic Algorithm to find f⁻¹(x):

  1. Set the given function equation equal to y: y = f(x).
  2. Interchange the roles of variables: Swap all instances of x and y.
  3. Rearrange the new algebraic equation to isolate and solve strictly for y.
  4. Replace the isolated y with the notation f⁻¹(x).