7.4 Composition and Inverse Functions

Function composition and inverses are powerful tools in mathematics. They allow us to combine and reverse functions, expanding our problem-solving toolkit. Understanding these concepts is crucial for tackling complex mathematical relationships.

Composition lets us chain functions together, while inverses undo a function's effect. These ideas are fundamental in many areas of math and science, from calculus to computer programming. Mastering them opens doors to advanced mathematical thinking.

Function Composition

Composition of functions

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• Function composition combines two functions applying one to output of another $(f \circ g)(x) = [f(g(x))](https://www.fiveableKeyTerm:f(g(x)))$
• Domain of composite function contains elements from g's domain mapping to f's domain $\{x \in \text{Dom}(g) : g(x) \in \text{Dom}(f)\}$
• Codomain of composite function forms subset of f's codomain $\text{Cod}(f \circ g) \subseteq \text{Cod}(f)$

Finding composite functions

1. Identify inner (g) and outer (f) functions
2. Substitute g(x) into f(x)
3. Simplify resulting expression

• Compose various function types (polynomials, trigonometric, exponential, logarithmic)
• Composition not commutative $f \circ g \neq g \circ f$ (generally)

Function Inverses

Inverse functions

• Inverse function reverses effect of original function $f^{-1}(y) = x$ if and only if $f(x) = y$
• Function must be bijective (one-to-one and onto) to have inverse
• Horizontal line test checks one-to-one property graphically
• Vertical line test verifies if relation is a function

Inverses of bijective functions

1. Replace $f(x)$ with $y$
2. Interchange $x$ and $y$
3. Solve for $y$
4. Replace $y$ with $f^{-1}(x)$

• Domain of $f^{-1}$ is codomain of $f$, codomain of $f^{-1}$ is domain of $f$
• Inverse functions graphically represented by reflection over $y = x$ line

Properties of compositions and inverses

• Function composition with inverse yields identity $(f \circ f^{-1})(x) = x$ and $(f^{-1} \circ f)(x) = x$
• Inverse of composite function $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$
• Bijective functions have unique inverses
• Inverse of bijective function is also bijective
• Composition preserves injectivity and surjectivity for both functions