{"id":7952,"date":"2026-02-15T22:01:25","date_gmt":"2026-02-15T16:31:25","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7952"},"modified":"2026-02-15T22:01:34","modified_gmt":"2026-02-15T16:31:34","slug":"how-to-prove-one-one-functions-with-examples","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/how-to-prove-one-one-functions-with-examples\/","title":{"rendered":"How to prove one-one functions with examples"},"content":{"rendered":"\n

This post we will see given a function, how to check if it is one-one function or not. Lets first start with the definition of the one-one functions<\/p>\n\n\n\n

Definition of one-one functions<\/h2>\n\n\n\n

A function f: A-> B is said to be one-one <\/p>\n\n\n\n

if, and only if, for all elements a1<\/sub> and a2<\/sub> in A,

if f(a1<\/sub>) = f(a2<\/sub>), then a1<\/sub> = a2<\/sub><\/p>\n\n\n\n

How to prove one-one function<\/h2>\n\n\n\n

To prove that a function is one-to-one (also known as injective), you need to show that no two distinct inputs have the same output. Here’s the general method for proving that a function f: A -> B is one-to-one:<\/p>\n\n\n\n

Method 1<\/strong><\/p>\n\n\n\n

    \n
  1. Assume that $f(x_1) = f(x_2)$ for some x1 and x2 in the domain A.<\/li>\n\n\n\n
  2. Use this assumption to show that $x_1 = x_2$.For example, you might start by assuming that f(x1) = f(x2), and then subtracting one equation from the other:f(x1) – f(x2) = 0 and then proving it<\/li>\n<\/ol>\n\n\n\n

    Example<\/p>\n\n\n\n

    f: R -> R given as f(x) = 2x +3<\/p>\n\n\n\n

    So,<\/p>\n\n\n\n

    $f(x_1)=f(x_2)$
    $2x_1 +3= 2x_2 +3$
    or<\/p>\n\n\n\n

    $x_1=x_2$<\/p>\n\n\n\n

    Conclude that the function is one-to-one. Since you have shown that no two distinct inputs have the same output, you can conclude that the function is one-to-one<\/p>\n\n\n\n

    Method 2<\/strong><\/p>\n\n\n\n

    If you know the graph or you can draw the graph easily, then you can use Horizontal line test to determine if it is one-one function or not<\/p>\n\n\n\n

    if a horizontal line intersects the graph at two points then it is not an one-one function. This is called the Horizontal line test<\/strong>. If all horizontal lines intersect a curve at most once then the curve represents a one-one function.<\/p>\n\n\n\n

    Example<\/p>\n\n\n\n

    The graph of the function f:R ->R f(x)=x(4-x) as<\/p>\n\n\n\n

    \"horizontal<\/a><\/figure>\n\n\n\n

    So this is not one-one function<\/p>\n\n\n\n

    Method <\/strong>3<\/p>\n\n\n\n

    Take counter examples<\/p>\n\n\n\n

    Example Let function f : R -> R, defined as $f(x) = x^2$<\/p>\n\n\n\n

    Solution<\/p>\n\n\n\n

    We can try to counter examples to check<\/p>\n\n\n\n

    here we can see that f(1) =1 ,f(-1) =1<\/p>\n\n\n\n

    Hence it is not a one-one function<\/p>\n\n\n\n

    Method 4<\/strong><\/p>\n\n\n\n

    Strictly increasing or decreasing functions are one-one<\/p>\n\n\n\n

    for f(x) if $\\frac {d f(x)}{dx} > 0$, Then f(x) is strictly increasing. So we can find the derivative and then determine if it is one-one or not
    for f(x) if $\\frac {d f(x)}{dx} < 0$, Then f(x) is strictly decreasing. So we can find the derivative and then determine if it is one-one or not<\/p>\n\n\n\n

    Example<\/p>\n\n\n\n

    Let f:[2,$\\infty$) -> R, $f(x)=x^3 -2x^2 +1$<\/p>\n\n\n\n

    $\\frac {d f(x)}{dx} = 3x^2 -4x= x(3x-4) > 0 , x> 2$<\/p>\n\n\n\n

    Hence f(x) is one -one function<\/p>\n\n\n\n

    Solved Examples<\/h2>\n\n\n\n

    Example 1<\/strong><\/p>\n\n\n\n

    f: R -> R such that $f(x)=\\frac {x}{x^2 + 1}$. Check if this one-one<\/p>\n\n\n\n

    Solution<\/strong><\/p>\n\n\n\n

    $f(x_1)=f(x_2)$<\/p>\n\n\n\n

    $\\frac {x_1}{x_1^2 + 1}=\\frac {x_2}{x_2^2 + 1}$<\/p>\n\n\n\n

    $(x_2 -x_1)(x_1 x_2 -1)=0$<\/p>\n\n\n\n

    So either $x_1=x_2$
    or
    $x_1 x_2 =1$
    here since x belongs to R, we can have $x_1=2$ and $x_2=1\/2$ as values satisfying this<\/p>\n\n\n\n

    Also f(2)=2\/5
    and f(1\/2)=2\/5<\/p>\n\n\n\n

    Hence this is not one-one functions<\/p>\n","protected":false},"excerpt":{"rendered":"

    This post we will see given a function, how to check if it is one-one function or not. Lets first start with the definition of the one-one functions Definition of one-one functions A function f: A-> B is said to be one-one if, and only if, for all elements a1 and a2 in A, if […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-7952","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nHow to prove one-one functions with examples - physicscatalyst's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/how-to-prove-one-one-functions-with-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How to prove one-one functions with examples - physicscatalyst's Blog\" \/>\n<meta property=\"og:description\" content=\"This post we will see given a function, how to check if it is one-one function or not. 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