Glossary

3.3 Functions as special relations

3 min read•august 7, 2024

Functions are special relations that map each element from one set to exactly one element in another set. They're crucial in math, defining relationships between inputs and outputs. This topic explores basics, properties, and advanced concepts.

We'll dive into function components like and , various notations, and key properties such as one-to-one correspondence. We'll also look at images, preimages, and inverse functions, building a solid foundation for understanding these essential mathematical tools.

Function Basics

Defining Functions and Their Components

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  • Function maps each element from one set (domain) to exactly one element in another set ()
  • Domain consists of all possible input values for the function
    • Elements in the domain are called arguments or inputs
  • Codomain includes all potential output values that the function could produce
    • Elements in the codomain are called outputs
  • Range represents the actual output values produced by the function
    • Range is a subset of the codomain
    • Range only includes elements that are mapped to by at least one element in the domain

Notations and Representations

  • Functions can be represented using various notations
    • f:X→Yf: X \to Y denotes a function ff from set XX to set YY
    • [f(x)](https://www.fiveableKeyTerm:f(x))=y[f(x)](https://www.fiveableKeyTerm:f(x)) = y indicates that the function ff maps the input xx to the output yy
  • Functions can also be represented graphically
    • Each point on the graph corresponds to an input-output pair (x,f(x))(x, f(x))
  • Functions can be described using tables or ordered pairs
    • Each row in the table or each represents an input-output relationship

Function Properties

One-to-One Correspondence

  • One-to-one correspondence () occurs when each element in the codomain is mapped to by at most one element in the domain
    • No two distinct elements in the domain map to the same element in the codomain
  • One-to-one functions have the property that if f(a)=f(b)f(a) = f(b), then a=ba = b
  • can be used to determine if a function is one-to-one
    • If any horizontal line intersects the graph of the function more than once, the function is not one-to-one

Image and Preimage

  • of an element xx under a function ff is the output value f(x)f(x)
    • Im(x)=f(x)\text{Im}(x) = f(x)
  • (inverse image) of an element yy in the codomain is the set of all elements in the domain that map to yy
    • Pre(y)={x∈X:f(x)=y}\text{Pre}(y) = \{x \in X : f(x) = y\}
  • For a , each element in the codomain has at most one preimage

Advanced Function Concepts

Inverse Functions

  • f−1f^{-1} "undoes" the original function ff
    • If f(a)=bf(a) = b, then f−1(b)=af^{-1}(b) = a
  • For a function f:X→Yf: X \to Y to have an inverse, it must be a bijection (one-to-one and onto)
    • One-to-one ensures each element in the codomain has at most one preimage
    • Onto ensures each element in the codomain has at least one preimage
  • Inverse function can be found by swapping xx and yy in the original function and solving for yy
    • Example: If f(x)=2x+1f(x) = 2x + 1, then f−1(x)=x−12f^{-1}(x) = \frac{x-1}{2}
  • Graphs of inverse functions are reflections of each other across the line y=xy = x

Key Terms to Review (22)

Bijective Function: A bijective function is a type of function that establishes a one-to-one correspondence between elements of two sets, meaning that every element in the first set is paired with a unique element in the second set and vice versa. This characteristic ensures that both the function is injective (no two elements from the first set map to the same element in the second) and surjective (every element in the second set is an image of at least one element from the first). Understanding bijective functions is crucial because they allow for effective comparisons of set sizes and play a fundamental role in various branches of mathematics, including topology and computer science.
Codomain: The codomain of a function is the set that contains all the possible output values that a function can produce based on its input values. It plays a crucial role in defining functions as special relations, as it establishes a boundary for the outputs and helps distinguish between the range (actual outputs) and potential outputs.
Composition of Functions: The composition of functions is a mathematical operation that takes two functions, say f and g, and combines them to create a new function, denoted as (f ∘ g)(x) = f(g(x)). This process allows you to apply one function to the output of another, effectively chaining their operations. This concept illustrates how functions can interact with each other, showcasing relationships between different mappings.
Constant function: A constant function is a type of function where the output value remains the same regardless of the input value. This means that for any input in the domain, the function produces a single fixed value, making it a simple yet essential concept in understanding how functions behave as special relations.
Domain: The domain of a function refers to the complete set of possible values that can be input into the function. It connects directly to the concept of relations and mappings, establishing the starting point for any function. Each element in the domain is paired with an output in the codomain, ensuring that every input is accounted for in the function's definition.
F: x → y: In set theory, the notation f: x → y describes a function f that maps elements from a set x (the domain) to elements in a set y (the codomain). This notation emphasizes the relationship where each input from set x is paired with exactly one output in set y, establishing a clear and consistent association between the two sets.
F(x): In mathematics, the notation f(x) represents a function, which is a special type of relation that assigns each input from a set, called the domain, to exactly one output in another set known as the codomain. This notation succinctly expresses how the function transforms inputs into outputs and is a fundamental concept when discussing relationships between variables in mathematics.
Function: A function is a special type of relation between two sets that assigns exactly one element from the second set to each element of the first set. This unique pairing ensures that every input has a single corresponding output, distinguishing functions from general relations. Functions can be represented in various ways, including equations, graphs, or tables, and they play a crucial role in various fields such as mathematics and computer science.
Graph of a function: The graph of a function is a visual representation of the relationship between the input values (or independent variable) and the output values (or dependent variable) of that function. Each point on the graph corresponds to an ordered pair, where the x-coordinate represents an input value and the y-coordinate represents the resulting output value. This visual representation allows us to easily analyze and interpret the behavior of functions, including their trends, intercepts, and continuity.
Horizontal Line Test: The horizontal line test is a method used to determine if a function is one-to-one, meaning that each output value corresponds to exactly one input value. If any horizontal line drawn through the graph of the function intersects the curve at more than one point, the function fails the test and is not one-to-one. This test is crucial for understanding functions, particularly in relation to their invertibility and uniqueness of outputs.
Image: In mathematics, particularly in the context of functions, the image of a set is the collection of all outputs that can be obtained from a specific function when applied to that set. This means that if you have a function mapping inputs to outputs, the image consists of all the outputs for the inputs taken from a particular subset of the domain.
Injective Function: An injective function, also known as a one-to-one function, is a type of function where every element in the domain maps to a unique element in the codomain. This means that if two inputs are different, their outputs must also be different, ensuring that no two elements in the domain share the same image. Understanding injective functions is crucial for exploring properties of relations and functions, particularly when distinguishing between different types of mappings, such as surjective and bijective functions, and analyzing their implications in set theory and mathematical structures.
Inverse Function: An inverse function is a function that reverses the effect of another function, meaning if a function $$f$$ takes an input $$x$$ and gives an output $$y$$, then the inverse function $$f^{-1}$$ takes that output $$y$$ and returns the original input $$x$$. This relationship highlights the special nature of functions as mappings between sets, where the original function and its inverse can be seen as two sides of the same coin. Understanding inverse functions also involves concepts such as injective and surjective functions, which determine whether a unique inverse exists.
Linear Function: A linear function is a specific type of function that creates a straight line when graphed on a coordinate plane. It is defined by the equation of the form $$f(x) = mx + b$$, where $$m$$ represents the slope and $$b$$ is the y-intercept. This straightforward relationship makes linear functions important in understanding how functions can represent constant rates of change.
One-to-One Function: A one-to-one function is a type of function where each input value corresponds to a unique output value, meaning no two different inputs produce the same output. This characteristic ensures that the function does not map multiple elements from its domain to a single element in its range, making it easier to determine the invertibility of the function. Understanding one-to-one functions is essential as it highlights how different elements relate to each other within functions, revealing their special properties.
Onto function: An onto function, also known as a surjective function, is a type of function where every element in the codomain has at least one pre-image in the domain. This means that the function covers the entire codomain, ensuring that no element is left unmatched. Understanding onto functions helps in studying the relationships between sets and how functions behave as special types of relations.
Ordered Pair: An ordered pair is a fundamental concept in set theory, defined as a pair of elements in which the order matters, typically denoted as (a, b). This means that (a, b) is different from (b, a) unless a and b are the same. Ordered pairs are crucial for establishing relationships between elements, forming the basis for concepts like binary relations and functions, and they also play a vital role in the construction of Cartesian products.
Piecewise function: A piecewise function is a type of function that is defined by multiple sub-functions, each of which applies to a specific interval or condition within the overall domain. These functions are used to describe situations where a single formula cannot capture the behavior of the function across its entire domain, allowing for different rules to be applied to different parts of the input values. This flexibility makes piecewise functions particularly useful in modeling real-world scenarios that involve distinct behaviors in different ranges of input.
Preimage: A preimage is the set of all elements in the domain of a function that map to a specific element in the codomain. In simpler terms, if you have a function that takes inputs and produces outputs, the preimage of an output is all the inputs that could produce that particular output. Understanding preimages helps clarify how functions relate different sets, emphasizing the idea that multiple inputs can lead to the same output.
Quadratic function: A quadratic function is a type of polynomial function of degree two, typically expressed in the form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a eq 0$$. This function creates a parabolic graph that opens either upward or downward, depending on the sign of the leading coefficient $$a$$. Quadratic functions have unique features like vertex, axis of symmetry, and roots that provide valuable information about their behavior and characteristics.
Range: The range of a function refers to the set of all possible output values it can produce based on its inputs. It connects closely with the concept of a function as a special type of relation, where each input is associated with one specific output, thereby highlighting the limits of output values. Additionally, understanding how the range relates to Cartesian products and ordered pairs allows for a deeper grasp of how values interact within a defined set, emphasizing the importance of both input and output relationships.
Vertical Line Test: The vertical line test is a method used to determine if a relation is a function by checking if any vertical line intersects the graph of the relation at more than one point. If such an intersection occurs, the relation does not represent a function since a function must assign exactly one output for each input. This test is crucial in visualizing and understanding the behavior of functions as special relations.
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