Function

Composite Function

If functions f: A→B and g: B→C are defined, and a relation A→C is also defined as a function, then the function A→C is called the composite function of f and g. This is denoted in the form g∘f: A→C or g∘f (x) or g∘f or gf(x) or gf.

Note: g∘f(x) = g(f(x))  → Read as: "g of f of x"

Step-by-Step Method to Find g∘f(x):

1. Start with the inner function: f(x)

2. Substitute f(x) into g(x) wherever you see "x"

3. Simplify the expression

Composite Function Rules:

1. Order matters: gf(x) ≠ fg(x) usually

2. gf(x) means: apply f FIRST, then g

3. Domain of gf = Domain of f (where f(x) is in domain of g)

Inverse Function

When the domain and range of any function f(x) are interchanged and the function is defined, then it is called the inverse function of f(x). The inverse function of f(x) is written as f−1(x) or f−1 and read as 'inverse function of f '. If the function is y = f(x), then for the inverse function x and y interchange their roles. That is, if x = f(y), then y = f−1(x).

Inverse Function Rules

• f⁻¹ exists ONLY if f is one-to-one AND onto

• f(f⁻¹(x)) = x  and  f⁻¹(f(x)) = x

• Graph of f⁻¹ is reflection of f across y = x line