Function operations and composition are powerful tools in algebra. They allow us to combine existing functions to create new ones, opening up a world of mathematical possibilities. By understanding how to add, subtract, multiply, and divide functions, we can model complex relationships.

Composition takes this a step further, letting us chain functions together. This mirrors real-world processes where one action's output becomes another's input. Mastering these concepts helps us tackle more advanced math and solve practical problems in various fields.

Function Operations and Composition

Combining functions algebraically

• Addition combines two functions $f$ and $g$ by adding their outputs for each input value $(f + g)(x) = f(x) + g(x)$ (sum of functions)
  • Domain of the sum function is the intersection of the domains of $f$ and $g$ because both functions must be defined at a given input for their sum to be defined
• Subtraction combines two functions $f$ and $g$ by subtracting the output of $g$ from the output of $f$ for each input value $(f - g)(x) = f(x) - g(x)$ (difference of functions)
  • Domain of the difference function is the intersection of the domains of $f$ and $g$ because both functions must be defined at a given input for their difference to be defined
• Multiplication combines two functions $f$ and $g$ by multiplying their outputs for each input value $(f \cdot g)(x) = f(x) \cdot g(x)$ (product of functions)
  • Domain of the product function is the intersection of the domains of $f$ and $g$ because both functions must be defined at a given input for their product to be defined
• Division combines two functions $f$ and $g$ by dividing the output of $f$ by the output of $g$ for each input value $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$ (quotient of functions)
  • Domain of the quotient function is the intersection of the domains of $f$ and $g$, excluding values where $g(x) = 0$ because division by zero is undefined

Creation of composite functions

• Composition creates a new function by applying one function $g$ to the input $x$, then applying another function $f$ to the result $(f \circ g)(x) = f(g(x))$ ("$f$ composed with $g$ of $x$")
  • Evaluate the inner function $g$ first, then use its output as the input for the outer function $f$ (order of operations)
• Order of composition matters because $(f \circ g)(x)$ is not always equal to $(g \circ f)(x)$ (non-commutative property)
• To evaluate a composite function $(f \circ g)(a)$ at a specific input $a$:
  1. Calculate the output of the inner function $g(a)$
  2. Use the output $g(a)$ as the input for the outer function $f$ and calculate $f(g(a))$
• Function notation is used to represent composite functions, such as $(f \circ g)(x)$ or $f(g(x))$

Domain of composite functions

• Domain of a composite function $(f \circ g)$ consists of all $x$ values in the domain of the inner function $g$ such that the output $g(x)$ is in the domain of the outer function $f$
• Steps to determine the domain of $(f \circ g)$:
  1. Find the domain of the inner function $g$
  2. Determine the values of $x$ for which the output $g(x)$ is in the domain of the outer function $f$
  3. Take the intersection of these two sets to obtain the domain of $(f \circ g)$
• The range of a composite function is the set of all possible output values produced by $(f \circ g)(x)$

Components of composite functions

• In a composite function $(f \circ g)(x)$, the outer function $f$ is applied last and the inner function $g$ is applied first
• To identify the component functions:
  • The outermost function is $f$ (applied last)
  • The innermost function is $g$ (applied first)

Applications of function composition

• Function composition models real-world situations where the output of one process becomes the input of another (chaining processes)
  • Converting units of measurement (inches to feet to meters)
    • $f$: inches to feet
    • $g$: feet to meters
    • $(f \circ g)$: inches directly to meters
  • Calculating profit based on revenue and expenses (revenue minus expenses)
    • $f$: revenue from units sold
    • $g$: expenses from units produced
    • $(g \circ f)$: profit by determining revenue first, then subtracting expenses
  • Determining final course grades (raw scores to percentages to letter grades)
    • $f$: raw scores to percentages
    • $g$: percentages to letter grades
    • $(g \circ f)$: letter grades directly from raw scores

Special Functions and Relationships

• One-to-one functions have a unique output for each unique input, allowing for the creation of inverse functions
• An inverse function "undoes" the operation of the original function, mapping the range back to the domain
• The identity function $f(x) = x$ is a special case where the input equals the output, often used in function composition