Glossary

3.2 Types of Functions: Injective, Surjective, and Bijective

3 min read•august 12, 2024

Functions are the backbone of mathematics, connecting inputs to outputs in fascinating ways. In this section, we'll explore three key types: injective, surjective, and bijective functions. Each has unique properties that shape how elements map between sets.

Understanding these function types is crucial for grasping more complex mathematical concepts. We'll dive into their definitions, characteristics, and real-world applications, building a solid foundation for future mathematical exploration.

Function Types

Injective Functions

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  • maps distinct elements of the domain to distinct elements in the codomain
  • Also known as
  • Each element in the codomain corresponds to at most one element in the domain
  • Mathematically expressed as: for all a and b in the domain, if f(a)=f(b)f(a) = f(b), then a=ba = b
  • Graphically represented by a curve that never intersects a horizontal line more than once
  • Ensures uniqueness in the process
  • Can be reversed to create an
  • Examples include:
    • The function f(x)=2xf(x) = 2x (multiplies each input by 2)
    • The function f(x)=x3f(x) = x^3 (cubes each input)

Surjective Functions

  • maps every element of the codomain to at least one element in the domain
  • Also known as
  • Range of the function equals its codomain
  • Mathematically expressed as: for every y in the codomain, there exists an x in the domain such that f(x)=yf(x) = y
  • Graphically represented by a curve that covers the entire range of the y-axis within the codomain
  • Ensures that all elements in the codomain are "reached" by the function
  • May have multiple elements in the domain mapping to the same element in the codomain
  • Examples include:
    • The function f(x)=x2f(x) = x^2 for x≥0x \geq 0 and codomain [0,∞)[0, \infty) (maps non-negative real numbers to non-negative real numbers)
    • The function f(x)=exf(x) = e^x with codomain (0,∞)(0, \infty) (maps real numbers to positive real numbers)

Bijective Functions

  • combines properties of both injective and surjective functions
  • Also known as one-to-one correspondence
  • Each element in the codomain corresponds to exactly one element in the domain
  • Mathematically expressed as: the function is both injective and surjective
  • Graphically represented by a curve that passes the and covers the entire codomain
  • Creates a perfect pairing between elements of the domain and codomain
  • Always has an inverse function
  • Useful in establishing equivalence between sets
  • Examples include:
    • The function f(x)=3x+2f(x) = 3x + 2 (linear function with non-zero slope)
    • The function f(x)=exf(x) = e^x with codomain (0,∞)(0, \infty) (exponential function with positive range)

Testing and Properties

Graphical Tests for Function Types

  • Horizontal line test determines injectivity of a function
  • Passing horizontal line test indicates injective function
  • Line intersects graph at most once for injective functions
  • Failing horizontal line test indicates non-injective function
  • Line intersects graph more than once for non-injective functions
  • determines if a relation is a function
  • Passing vertical line test confirms the relation is a function
  • Line intersects graph at most once for functions
  • Failing vertical line test indicates the relation is not a function
  • Line intersects graph more than once for non-functions

Function Images and Preimages

  • Image refers to the set of all output values produced by a function
  • Denoted as f(A)f(A) for a subset A of the domain
  • Mathematically expressed as: f(A)={y∈Y:y=f(x) for some x∈A}f(A) = \{y \in Y : y = f(x) \text{ for some } x \in A\}
  • Represents the "result" of applying the function to a set of inputs
  • Can be smaller than, equal to, or larger than the original set A
  • Preimage refers to the set of all input values that produce a specific output
  • Also known as inverse image
  • Denoted as f−1(B)f^{-1}(B) for a subset B of the codomain
  • Mathematically expressed as: f−1(B)={x∈X:f(x)∈B}f^{-1}(B) = \{x \in X : f(x) \in B\}
  • Represents all elements in the domain that map to a given subset of the codomain
  • Can be empty, contain a single element, or multiple elements
  • Examples:
    • For f(x)=x2f(x) = x^2, the image of [−1,1][-1, 1] is [0,1][0, 1]
    • For f(x)=sin⁡(x)f(x) = \sin(x), the preimage of {1}\{1\} is {π2+2πn:n∈Z}\{\frac{\pi}{2} + 2\pi n : n \in \mathbb{Z}\}

Key Terms to Review (17)

Bijective function: A bijective function is a type of function that establishes a one-to-one correspondence between elements in its domain and elements in its codomain, meaning that each element in the domain maps to exactly one unique element in the codomain, and vice versa. This property implies that a bijective function is both injective (one-to-one) and surjective (onto), ensuring that no two elements in the domain map to the same element in the codomain and every element in the codomain is an image of some element in the domain. The significance of bijective functions lies in their ability to create invertible mappings, making them essential in various mathematical applications, including set theory and problem-solving techniques.
Cardinality: Cardinality refers to the number of elements in a set, which helps to measure the size of the set. Understanding cardinality is crucial when comparing the sizes of different sets, whether they are finite, countably infinite, or uncountably infinite. It also plays a significant role in understanding relationships between sets and their properties, as well as how functions map elements from one set to another.
F: a → b: In mathematics, the notation 'f: a → b' represents a function 'f' that maps elements from set 'a' (the domain) to set 'b' (the codomain). This mapping implies that each element in 'a' is associated with exactly one element in 'b', establishing a relationship between the two sets. Understanding this notation is fundamental when analyzing different types of functions, such as injective, surjective, and bijective functions, which further classify how the elements in these sets interact with each other.
F(x) = 2x is injective: The function f(x) = 2x is injective, meaning that for every pair of distinct inputs, the outputs are also distinct. This property indicates that no two different values of x will yield the same value of f(x), establishing a one-to-one correspondence between elements in the domain and range. Injectivity is a fundamental characteristic of functions, differentiating them based on how they map inputs to outputs.
F(x) = 3x + 2 is bijective: A function is bijective if it is both injective and surjective, meaning every element in the codomain is mapped to by exactly one element in the domain. For the function f(x) = 3x + 2, it satisfies both properties, ensuring that each input corresponds to a unique output and that every possible output is achieved from some input.
F(x) = x^2 for x ≥ 0 is surjective: The function f(x) = x^2 for x ≥ 0 is called surjective because every element in its codomain has a corresponding element in its domain. A function is surjective if for every possible output value, there exists at least one input value that produces that output. In this case, the function maps non-negative real numbers to non-negative real numbers, meaning that every non-negative real number can be achieved as an output by choosing an appropriate non-negative input.
F^-1: The notation f^-1 represents the inverse function of a given function f. An inverse function essentially reverses the operation of the original function, meaning that if you apply f to an input x and then apply f^-1 to the result, you will get back x. This concept is closely tied to the ideas of injective, surjective, and bijective functions, as only bijective functions have an inverse that is also a function.
Function composition: Function composition is the process of combining two functions to create a new function, where the output of one function becomes the input of another. This operation allows for the evaluation of complex relationships between variables and can help simplify calculations by breaking them down into smaller, manageable parts. Understanding function composition is crucial for exploring how functions interact with one another, which relates directly to definitions, types of functions, and their inverses.
Horizontal line test: The horizontal line test is a method used to determine if a function is injective (one-to-one). If any horizontal line intersects the graph of the function at more than one point, the function fails the test and is not injective. This concept is essential for understanding different types of functions, particularly injective, surjective, and bijective functions.
If f is bijective, then f has an inverse: A function f is considered bijective if it is both injective (one-to-one) and surjective (onto). When a function is bijective, every element in the codomain has a unique pre-image in the domain, allowing for the construction of an inverse function that effectively 'reverses' the mapping of f. This relationship is crucial in understanding how functions can be inverted and highlights the significance of bijectivity in function analysis.
Injective Function: An injective function, or one-to-one function, is a type of function where each element in the domain maps to a unique element in the codomain. This means that no two different inputs produce the same output, ensuring that every value of the output corresponds to only one value of the input. Understanding injective functions is essential for distinguishing them from other types of functions, such as surjective and bijective functions, which have different properties related to the relationships between their sets.
Inverse function: An inverse function is a function that reverses the operation of the original function, meaning if the original function takes an input 'x' and produces an output 'y', the inverse function takes 'y' back to 'x'. This concept is crucial in understanding how functions interact and provides insights into the relationship between sets of inputs and outputs, particularly when discussing problem-solving strategies and the classification of functions.
Mapping: Mapping refers to a relationship or correspondence between two sets where each element of the first set is associated with an element of the second set. This concept is fundamental in mathematics as it allows us to understand how elements from one group can relate to elements in another group, creating a structured way to analyze data and functions. Mappings can take various forms, such as functions, and are essential in discussing power sets and Cartesian products as they define how elements from these sets interact.
One-to-one function: A one-to-one function, or injective function, is a type of function where each element in the domain maps to a unique element in the codomain, meaning no two different inputs produce the same output. This concept is crucial for understanding how functions relate elements in sets and plays a significant role in various mathematical contexts, including set theory and function types, as it helps establish the uniqueness of solutions and relationships between different mathematical entities.
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 element from the domain mapping to it. This means that the range of the function is equal to its codomain, ensuring that no part of the codomain is left unmapped. Understanding onto functions is essential for grasping various concepts related to sets and functions, including how they relate to other types of functions like injective and bijective functions.
Surjective Function: A surjective function, also known as an onto function, is a type of mapping where every element in the target set has at least one pre-image in the domain. This means that the function covers the entire range of possible outputs, ensuring that no element in the codomain is left unpaired with an input from the domain. Surjective functions play a vital role in understanding the relationships between different sets and are closely related to concepts like injective and bijective functions.
Vertical Line Test: The vertical line test is a visual method used to determine if a curve is a graph of a function. If any vertical line drawn through the graph intersects it at more than one point, then the relation represented by the graph is not a function. This test is crucial for distinguishing functions from non-functions and plays an important role in understanding function notation and types.
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