one-to-one function

(noun)

A function that never maps distinct elements of its domain to the same element of its range. 

Related Terms

Examples of one-to-one function in the following topics:

  • One-to-One Functions

    • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
    • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain.
    • One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.  
    • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  
    • If a horizontal line can go through two or more points on the function's graph then the function is NOT one-to-one.

  • Inverse Trigonometric Functions

    • For a one-to-one function, if $f(a) = b$, then an inverse function would satisfy $f^{-1}(b) = a$.
    • However, the sine, cosine, and tangent functions are not one-to-one functions.
    • In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods.
    • As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.
    • Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible.

  • Inverse Functions

    • Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs—the domain—to a set of outputs—the range.
    • Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range $Y$, in which case the inverse relation is the inverse function.
    • For this rule to be applicable, each element $y \in Y$ must correspond to no more than one $x \in X$; a function $f$ with this property is called one-to-one, information-preserving, or an injection.
    • To find the inverse of this function, undo each of the operations on the $x$ side of the equation one at a time.
    • We can check to see if this inverse "undoes" the original function by plugging that function in for $x$:

  • Functional Groups

    • Similarly, a functional group can be referred to as primary, secondary, or tertiary, depending on if it is attached to one, two, or three carbon atoms .
    • Functionalization refers to the addition of functional groups to a compound by chemical synthesis.
    • In materials science, functionalization is employed to achieve desired surface properties; functional groups can also be used to covalently link functional molecules to the surfaces of chemical devices.
    • They can be classified as primary, secondary, or tertiary, depending on how many carbon atoms the central carbon is attached to.
    • Define the term "functional group" as it applies to organic molecules

  • Inverse Functions

    • If $f$ is invertible, the function $g$ is unique; in other words, there is exactly one function $g$ satisfying this property (no more, no less).
    • For this rule to be applicable, for a function whose domain is the set $X$ and whose range is the set $Y$, each element $y \in Y$ must correspond to no more than one $x \in X$; a function $f$ with this property is called one-to-one, or information-preserving, or an injection.
    • Direct variation function are based on multiplication; $y=kx$.
    • The function $f(x)=x^2$ may or may not be invertible, depending on the domain.
    • Because $f$ maps $a$ to 3, the inverse $f^{-1}$ maps 3 back to $a$.

  • Increasing, Decreasing, and Constant Functions

    • The figure below shows examples of increasing and decreasing intervals on a function.
    • Look at the graph from left to right on the $x$-axis; the first part of the curve is decreasing from infinity to the $x$-value of $-1$ and then the curve increases.  
    • The curve increases on the interval from $-1$ to $1$ and then it decreases again to infinity.
    • The function $f(x)=x^3−12x$ is increasing on the $x$-axis from negative infinity to $-2$ and also from $2$ to positive infinity.  
    • The function is decreasing on on the interval: $ (−2, 2)$.  

  • Linear and Quadratic Functions

    • They are one of the simplest functional forms.
    • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
    • However, the term "linear function" is quite often loosely used to include affine functions of the form $f(x)=mx+b$.
    • The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis .
    • If the quadratic function is set equal to zero, then the result is a quadratic equation.

  • Introduction to Rational Functions

    • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
    • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
    • Factorizing the numerator and denominator of rational function helps to identify singularities of algebraic rational functions.
    • We can factor the denominator to find the singularities of the function:
    • However, for $x^2 + 2=0$ , $x^2$ would need to equal $-2$.

  • Functional Structure

    • An organization with a functional structure is divided based on functional areas, such as IT, finance, or marketing.
    • Some refer to these functional areas as "silos"—entities that are vertical and disconnected from each other.
    • A disadvantage of this structure is that the different functional groups may not communicate with one another, potentially decreasing flexibility and innovation.
    • Functional structures may also be susceptible to tunnel vision, with each function perceiving the organization only from within the frame of its own operation.
    • Smaller companies that require more adaptability and creativity may feel confined by the communicative and creative silos functional structures tend to produce.

  • Continuity

    • Otherwise, a function is said to be a "discontinuous function."
    • Continuity of functions is one of the core concepts of topology.
    • The function $f$ is continuous at some point $c$ of its domain if the limit of $f(x)$ as $x$ approaches $c$ through the domain of $f$ exists and is equal to $f(c)$.
    • the limit on the left-hand side of that equation has to exist, and
    • The function $f$ is said to be continuous if it is continuous at every point of its domain.