Inverse functions

Quiz 1

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Understanding Composite and Inverse Functions: Paraphrased Explanation and Key Formulas

Composite and inverse functions are essential concepts in mathematics that describe how functions combine and how one function can reverse the effect of another. This guide offers a rewritten explanation of these ideas with important formulas, optimized for mobile devices and Blogger compatibility.

1. Composite Functions

A composite function is created by applying one function to the output of another. If you have two functions, f and g, their composition is written as (f ∘ g)(x), which means:

(f ∘ g)(x) = f(g(x))

This indicates that you first calculate g(x), then plug that result into f.

Key facts to remember:

• The order matters: generally, (f ∘ g)(x) is not the same as (g ∘ f)(x).
• The composite function (f ∘ g) is defined only if the outputs of g fit within the inputs allowed for f.
• The domain of (f ∘ g) corresponds to the domain of g.

Properties of Composite Functions

• Associativity: For functions f, g, h, the composition satisfies f ∘ (g ∘ h) = (f ∘ g) ∘ h.
• Not Commutative: Usually, switching the order changes the result, so f ∘ g ≠ g ∘ f.
• Preservation of Injectivity and Surjectivity: Composing one-to-one functions results in a one-to-one function; composing onto functions results in an onto function.

Example

Given f(x) = 2x + 3 and g(x) = x², then:

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

For example, when x = 2, (f ∘ g)(2) = 2(4) + 3 = 11.

2. Inverse Functions

An inverse function reverses the effect of a given function. For a function f that maps elements from set X to set Y, the inverse function f⁻¹ maps elements back from Y to X. This inverse exists only if f is both one-to-one (injective) and onto (surjective), collectively called bijective.

f(f⁻¹(x)) = f⁻¹(f(x)) = x

The domain of f⁻¹ is the range of f, and its range is the domain of f.

Steps to Find an Inverse Function

1. Express the function as y = f(x).
2. Switch the variables x and y, so you write x = f(y).
3. Solve this equation for y in terms of x.
4. Rename y as f⁻¹(x), which is the inverse function.

Example

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

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

Inverse of a Composite Function

The inverse of a composite function (f ∘ g) is the composition of the inverses in reverse order:

(f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹

This means to find the inverse of (f ∘ g), first apply f⁻¹, then g⁻¹.

Summary Table of Important 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 Insights

• Only bijective functions have inverses.
• Always check domain and range restrictions when dealing with inverses.
• Composite functions are useful for modeling multi-step processes in mathematics.