Composite Function

Quiz 1

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Composite Functions and Inverse Functions: Theory and Formulas

In mathematics, composite functions and inverse functions are fundamental concepts that describe how functions combine and reverse each other's effects. This document provides a comprehensive theory and formulas about these topics, optimized for mobile viewing and compatible with Blogger.

1. Composite Functions

A composite function is formed when one function is applied to the result of another function. Given two functions f and g, their composition is denoted as (f ∘ g)(x) and defined as:

(f ∘ g)(x) = f(g(x))

This means you first apply g to x, then apply f to the result of g(x).

Important points:

• The order of composition matters: generally, (f ∘ g)(x) ≠ (g ∘ f)(x).
• The composite function (f ∘ g) is defined only if the range of g is a subset of the domain of f.
• The domain of (f ∘ g) is the domain of g.

Properties of Composite Functions

• Associative Property: For functions f, g, h, f ∘ (g ∘ h) = (f ∘ g) ∘ h.
• Non-Commutative: Usually, f ∘ g ≠ g ∘ f.
• One-to-One and Onto: The composition of one-to-one functions is one-to-one; the composition of onto functions is onto.

Example of Composite Function

Let f(x) = 2x + 3 and g(x) = x². Then,

(f ∘ g)(x) = f(g(x)) = f(x²) = 2x² + 3

If x = 2, then (f ∘ g)(2) = 2(2²) + 3 = 2(4) + 3 = 11.

2. Inverse Functions

An inverse function f⁻¹ of a function f reverses the effect of f. Formally, if f maps x to y, then f⁻¹ maps y back to x. The inverse function exists only if f is bijective (one-to-one and onto).

f(f⁻¹(x)) = f⁻¹(f(x)) = x

Here, the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

How to Find the Inverse Function Algebraically

1. Write the function as y = f(x).
2. Swap x and y to get x = f(y).
3. Solve this equation for y in terms of x.
4. Replace y by f⁻¹(x), which is the inverse function.

Example

Find the inverse of f(x) = 3x - 1.

y = 3x - 1
Swap x and y:
x = 3y - 1
Solve for y:
3y = x + 1
y = (x + 1) / 3
Therefore,
f⁻¹(x) = (x + 1) / 3

Inverse of Composite Functions

The inverse of a composite function (f ∘ g) is given by:

(f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹

This means you first apply the inverse of f, then the inverse of g.

Summary of Key Formulas

Concept	Formula / Definition
Composite Function	(f ∘ g)(x) = f(g(x))
Domain of Composite	dom(f ∘ g) = dom(g), if range(g) ⊆ dom(f)
Inverse Function	f(f⁻¹(x)) = f⁻¹(f(x)) = x
Inverse of Composite	(f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹

Additional Notes

• Not all functions have inverses; only bijective functions do.
• When finding inverses, always verify the domain and range restrictions.
• Composite functions are widely used in calculus, algebra, and applied mathematics to model complex operations.