one-to-one function

(noun)

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

Related Terms

  • inverse function

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)=bf(a) = bf(a)=b, then an inverse function would satisfy f−1(b)=af^{-1}(b) = af​−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.
  • Sine and Cosine as Functions

    • Recall that the sine and cosine functions relate real number values to the xxx- and yyy-coordinates of a point on the unit circle.
    • Notice how the sine values are positive between 000 and π\piπ, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π\piπ and 2π2\pi2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle.
    • As with the sine function, we can plots points to create a graph of the cosine function.
    • The points on the curve y=sinxy = \sin xy=sinx correspond to the values of the sine function on the unit circle.
    • The points on the curve y=cosxy = \cos xy=cosx correspond to the values of the cosine function on the unit circle.
  • 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 xxx-axis; the first part of the curve is decreasing from infinity to the xxx-value of −1-1−1 and then the curve increases.  
    • The curve increases on the interval from −1-1−1 to 111 and then it decreases again to infinity.
    • The function $f(x)=x^3−12x$ is increasing on the xxx-axis from negative infinity to −2-2−2 and also from 222 to positive infinity.  
    • The function is decreasing on on the interval: $ (−2, 2)$.  
  • Composition of Functions and Decomposing a Function

    • Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
    • The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
    • Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function.
    • In general, functional decompositions are worthwhile when there is a certain "sparseness" in the dependency structure; i.e. when constituent functions are found to depend on approximately disjointed sets of variables.
    • Practice functional composition by applying the rules of one function to the results of another function
  • Inverses of Composite Functions

    • A composite function represents, in one function, the results of an entire chain of dependent functions.
    • In mathematics, function composition is the application of one function to the results of another.
    • The functions ggg and fff are said to commute with each other if $g ∘ f = f ∘ g$.
    • A composite function represents in one function the results of an entire chain of dependent functions.
    • The entire chain of dependent functions are the ingredients, drinks, plates, etc., and the one composite function would be putting the entire chain together in order to calculate a larger population at the school.
  • Introduction to Rational Functions

    • A rational function is one such that f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}f(x)=​Q(x)​​P(x)​​, where Q(x)≠0Q(x) \neq 0Q(x)≠0; the domain of a rational function can be calculated.
    • Any function of one variable, xxx, 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 x2+2=0x^2 + 2=0x​2​​+2=0 , x2x^2x​2​​ would need to equal −2-2−2.
  • Translations

    • A translation of a function is a shift in one or more directions.
    • To translate a function vertically is to shift the function up or down.
    • To translate a function horizontally is the shift the function left or right.
    • When aaa is positive, the function is shifted to the right.  
    • When aaa is negative, the function is shifted to the left.  
  • Introduction to Exponential and Logarithmic Functions

    • Logarithmic functions and exponential functions are inverses of each other.
    • Lastly, as with all inverse functions, if we graph f(x)=logbxf(x)=log_{b}x f(x)=log​b​​x  and f−1(x)=bxf^{-1}(x)=b^{x}f​−1​​(x)=b​x​​ on the same plane, the graphs will be symmetric across the line y=xy=xy=x.
    • That is, if we fold the plane over the line y=xy=xy=x, the two curves will lie on each other.
    • Another way of thinking about this is that if we generate points on the curve of f(x)=logbxf(x)=log_{b}xf(x)=log​b​​x we can find the points on the curve of  f−1(x)=bxf^{-1}(x)=b^{x}f​−1​​(x)=b​x​​ by interchanging the xxx and yyy coordinates of the points.
    • Further, a point (t,u=bt)(t,u=b^t)(t,u=b​t​​) on the graph of f(x)f(x)f(x) yields a point (u,t=logbu)(u,t=log{_b}u)(u,t=log​b​​u) on the graph of the logarithm and vice versa.
  • Stretching and Shrinking

    • Multiplying the entire function f(x)f(x)f(x) by a constant greater than one causes all the yyy values of an equation to increase.
    • If the function f(x)f(x)f(x) is multiplied by a value less than one, all the yyy values of the equation will decrease, leading to a "shrunken" appearance in the vertical direction.  
    • If bbb is greater than one the function will undergo vertical stretching, and if bbb is less than one the function will undergo vertical shrinking.
    • If we want to vertically stretch the function by a factor of three, then the new function becomes:
    • If ccc is greater than one the function will undergo horizontal shrinking, and if ccc is less than one the function will undergo horizontal stretching.
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