1.4 Composition and inverse functions

Function composition and inverses are powerful tools for manipulating and analyzing mathematical relationships. By combining functions, we can model complex systems and processes, simplifying them into more manageable parts.

Inverse functions allow us to "undo" operations, solving for inputs given outputs. This concept is crucial in real-world applications, from calculating investment returns to analyzing physical phenomena like projectile motion.

Function Composition and Inverses

Composition of multiple functions

• Applies one function to the result of another function
• Denoted as $(f \circ g)(x) = f(g(x))$, read as "$f$ composed with $g$ of $x$"
• Evaluates the inner function $g(x)$ first, then applies the outer function $f$ to the result
• Can compose more than two functions by applying them in order from right to left $(f \circ g \circ h)(x) = f(g(h(x)))$
• Simplifies complex functions into a single function
• Models multi-step processes or systems (cost of production, demand at a given price)

Domain and range in composites

• Domain of $f(g(x))$ is the set of all $x$ values for which both $g(x)$ and $f(g(x))$ are defined
  • Subset of the domain of $g(x)$
• Find the domain by determining $x$ values that satisfy both the domain of $g(x)$ and the values of $g(x)$ that are in the domain of $f(x)$
• Range of $f(g(x))$ is the set of all possible output values of the composition
  • Subset of the range of $f(x)$
• Compositions can restrict the domain and range of the original functions

Inverse function identification

• Inverse function $f^{-1}(x)$ "undoes" the original function $f(x)$
  • If $f(a) = b$, then $f^{-1}(b) = a$
• Function has an inverse if and only if it is one-to-one (injective)
  • Each element in the codomain is mapped to by at most one element in the domain
• Steps to find the inverse:
  1. Replace $f(x)$ with $y$
  2. Swap $x$ and $y$
  3. Solve the equation for $y$
  4. Replace $y$ with $f^{-1}(x)$
• Domain of $f^{-1}(x)$ is the range of $f(x)$, and vice versa
• Composition of a function with its inverse results in the identity function $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$

Applications of composition and inverses

• Model multi-step processes or systems
  • Production cost for demanded items at a given price $(f \circ g)(x)$
  • Compound interest earned on an investment over time $(f \circ g)(t)$
• Solve for inputs given outputs or "reverse" a process
  • Time to reach a given height for an object thrown upward $f^{-1}(y)$
  • Original investment amount given the final balance and interest rate $f^{-1}(b)$
• Simplify complex functions by breaking them down into smaller, more manageable parts
• Analyze relationships between variables in real-world scenarios (price and demand, time and growth)