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3.4 Composition of Functions

4 min read•june 24, 2024

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

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Top images from around the web for Combining functions algebraically

  • Find domain and range from graphs | College Algebra View original

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

Creation of composite functions

  • Composition creates a new function by applying one function gg to the input xx, then applying another function ff to the result (f∘g)(x)=[f(g(x))](https://www.fiveableKeyTerm:f(g(x)))(f \circ g)(x) = [f(g(x))](https://www.fiveableKeyTerm:f(g(x))) ("ff composed with gg of xx")
    • Evaluate the gg first, then use its output as the input for the ff (order of operations)
  • Order of composition matters because (f∘g)(x)(f \circ g)(x) is not always equal to (g∘f)(x)(g \circ f)(x) ()
  • To evaluate a (f∘g)(a)(f \circ g)(a) at a specific input aa:
    1. Calculate the output of the inner function g(a)g(a)
    2. Use the output g(a)g(a) as the input for the outer function ff and calculate f(g(a))f(g(a))
  • is used to represent composite functions, such as (f∘g)(x)(f \circ g)(x) or f(g(x))f(g(x))

Domain of composite functions

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

Components of composite functions

  • In a composite function (f∘g)(x)(f \circ g)(x), the outer function ff is applied last and the inner function gg is applied first
  • To identify the component functions:
    • The outermost function is ff (applied last)
    • The innermost function is gg (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)
      • ff: inches to feet
      • gg: feet to meters
      • (f∘g)(f \circ g): inches directly to meters
    • Calculating profit based on revenue and expenses (revenue minus expenses)
      • ff: revenue from units sold
      • gg: expenses from units produced
      • (g∘f)(g \circ f): profit by determining revenue first, then subtracting expenses
    • Determining final course grades (raw scores to percentages to letter grades)
      • ff: raw scores to percentages
      • gg: percentages to letter grades
      • (g∘f)(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 "undoes" the operation of the original function, mapping the range back to the domain
  • The f(x)=xf(x) = x is a special case where the input equals the output, often used in function composition

Key Terms to Review (37)

Associative Property of Composition: The associative property of composition states that the order in which functions are composed does not affect the final result. In other words, the composition of functions is associative, meaning that the way the functions are grouped together does not change the overall outcome.
Chain Rule: The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It allows you to differentiate functions that are built up by combining multiple functions, by breaking down the composite function into its individual components.
Codomain: The codomain of a function is the set of all possible output values that the function can produce. It represents the full range of values that the function is capable of mapping its input values to.
Combining functions: Combining functions involves creating a new function by applying one function to the results of another. This process is also known as the composition of functions.
Commutative: A property of functions where the order in which two functions are composed does not affect the result. Mathematically, if $f$ and $g$ are commutative, then $f(g(x)) = g(f(x))$ for all $x$ in the domain.
Composite Function: A composite function is a new function created by combining two or more functions, where the output of one function becomes the input of the next function. It allows for the chaining of functions to perform more complex operations.
Composition of Functions (f ∘ g): The composition of two functions, denoted as f ∘ g, is a new function that is created by applying one function (g) to the input of another function (f). The resulting function represents the combined effect of applying both functions in succession.
Cost function: A cost function represents the cost of producing a certain number of goods or services as a function of the quantity produced. It is typically expressed in algebraic form and used to model economic behavior.
Decompose a composite function: To decompose a composite function means to break it down into two or more simpler functions whose composition results in the original function. It helps in understanding the underlying structure and behavior of complex functions.
Decreasing linear function: A decreasing linear function is a linear function where the value of the function decreases as the input increases. It has a negative slope.
Difference of Functions: The difference of functions is a mathematical operation that involves subtracting one function from another. This concept is particularly important in the context of function composition, as the difference between functions can provide insights into their relationship and behavior.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Domain of a composite function: The domain of a composite function is the set of all input values for which the composed function is defined. It must satisfy the domains of both the inner and outer functions.
Evaluation: Evaluation is the process of assessing or determining the value, quality, or importance of something, often in the context of a specific task or goal. It involves carefully examining and analyzing information to make judgments or decisions.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
F(g(x)): The term f(g(x)) represents the composition of two functions, where the inner function g(x) is evaluated first, and then the result is used as the input for the outer function f(x). This allows for the combination of two separate functions into a single, more complex function.
Function notation: Function notation is a way to represent functions in the form $f(x)$, where $f$ names the function and $x$ represents the input variable. It provides a concise and clear way to denote the output of a function given an input.
Function Notation: Function notation is a way of representing and working with functions, where the function is denoted by a letter or symbol, and the input values are placed within parentheses after the function name. This notation allows for the clear and concise representation of functional relationships, which is essential in understanding and manipulating functions in various mathematical contexts.
General form of a quadratic function: The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$. This representation is crucial for solving quadratic equations and analyzing their properties.
Identity Function: The identity function is a special type of function where the output value is always equal to the input value. It is a fundamental concept in mathematics that is particularly relevant in the study of function composition and linear functions.
Inner Function: An inner function is a function that is defined within another function, known as the outer function. Inner functions have access to variables and parameters of the outer function, as well as their own local variables and parameters. They provide a way to encapsulate and hide functionality, creating a more modular and organized code structure.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Linear Function: A linear function is a mathematical function that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, known as the slope, and can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
Non-Commutative Property: The non-commutative property refers to a mathematical operation where the order of the operands affects the result. This means that for a given operation, $a \circ b$ is not necessarily equal to $b \circ a$, unlike the commutative property where the order does not matter.
One-to-one function: A one-to-one function (injective function) is a function where each element of the domain maps to a unique element in the codomain. No two different elements in the domain map to the same element in the codomain.
One-to-One Function: A one-to-one function, also known as an injective function, is a function where each element in the domain is mapped to a unique element in the codomain. This means that for every input value, there is only one corresponding output value, and no two input values can be mapped to the same output value.
Outer Function: The outer function is a higher-order function that takes one or more functions as arguments and returns a new function. It is a fundamental concept in functional programming, allowing for the creation of flexible and reusable code by encapsulating and composing smaller, modular functions.
Product of Functions: The product of functions refers to the result of multiplying two or more functions together. This operation is fundamental in the study of composition of functions, as the product of functions is often used to represent and analyze the relationships between different functions.
Quadratic Function: A quadratic function is a polynomial function of degree two, where the highest exponent of the independent variable is two. Quadratic functions are widely used in various mathematical and scientific applications, including physics, engineering, and economics.
Quotient of Functions: The quotient of functions refers to the result of dividing one function by another function. It represents the ratio of the values of the two functions at corresponding points in their domains.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Restricted Domain: The restricted domain of a function refers to the limited range of input values for which the function is defined. It represents the subset of the domain where the function can be evaluated without resulting in undefined or invalid outputs.
Substitution: Substitution is the process of replacing one or more elements in an expression with other values or variables to simplify, evaluate, or solve the expression. This technique is commonly used in various mathematical contexts, including the composition of functions, verifying trigonometric identities, and solving trigonometric equations.
Sum of Functions: The sum of functions is the operation of adding two or more functions together, resulting in a new function that represents the combined behavior of the original functions. This concept is fundamental in understanding the composition of functions, as the sum of functions is a crucial component in the composition process.
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3.5 Transformation of Functions