{"id":34790,"date":"2023-10-08T21:12:01","date_gmt":"2023-10-08T21:12:01","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=34790"},"modified":"2024-02-05T16:19:26","modified_gmt":"2024-02-05T16:19:26","slug":"onto-function-definition-with-examples","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function","title":{"rendered":"Onto Function: Definition, Formula, Properties, Graph, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-56568edd-5484-4ba6-9370-dd4e0cdea8e3\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#0-what-is-an-onto-function>What Is an Onto Function?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#2-how-to-prove-a-function-is-onto>How to Prove a Function is Onto<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#3-number-of-onto-functions-formula>Number of Onto Functions Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#8-solved-examples-on-onto-function>Solved Examples on Onto Function<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#9-practice-problems-on-onto-function>Practice Problems on Onto Function<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function#10-frequently-asked-questions-about-the-onto-function>Frequently Asked Questions about the Onto Function<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-an-onto-function\">What Is an Onto Function?<\/h2>\n\n\n\n<p><strong>Onto function is also known as a surjective function. For an onto function, each element in the second set (co-domain) has a preimage.&nbsp;<\/strong><\/p>\n\n\n\n<p>When we define a function from set X to set Y, the set X containing input values is called the domain of the function. The set Y of possible output values is called the co-domain. The subset of the co-domain containing the actual outputs is called the range of the function.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"627\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/domain-co-domain-and-range-of-a-function-visual.png\" alt=\"Domain, co-domain, and range of a function visual\" class=\"wp-image-34794\" title=\"Domain, co-domain, and range of a function visual\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/domain-co-domain-and-range-of-a-function-visual.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/domain-co-domain-and-range-of-a-function-visual-297x300.png 297w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/domain-co-domain-and-range-of-a-function-visual-120x120.png 120w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>In case of onto functions, we have range = co-domain. It means that all the elements of the co-domain represent the range.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"461\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function.png\" alt=\"Onto function\" class=\"wp-image-34795\" title=\"Onto function\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function-300x223.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p>So, if a single element in Y is not mapped to an element of X, the function is not onto.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"500\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function-example-and-non-example.png\" alt=\"Onto function example and non-example\" class=\"wp-image-34797\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function-example-and-non-example.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/onto-function-example-and-non-example-300x242.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"1-onto-function-definition\">Onto Function Definition<\/h2>\n\n\n\n<p><strong>A function f : A <\/strong>\u2192 <strong>B is called an onto function if each element of B is mapped to at least one element of A. For every b \u2208<\/strong> <strong>B, there is a \u2208 A such that f(a) = b.<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"2-how-to-prove-a-function-is-onto\">How to Prove a Function is Onto<\/h2>\n\n\n\n<p>To show that a function f is an onto function, put y = f(x), and show that we can express x in terms of y for any y \u2208 B.<\/p>\n\n\n\n<p><strong>Example 1: <\/strong>f:R \u2192 R defined as f(x) = 2x<\/p>\n\n\n\n<p>y = 2x<\/p>\n\n\n\n<p>x $= \\frac{y}{2}$<\/p>\n\n\n\n<p>Thus, for every y R, we have $x = \\frac{y}{2} \u2208 R such that f(x) = y.&nbsp;<\/p>\n\n\n\n<p>Thus, f is onto.<\/p>\n\n\n\n<p><strong>Example 2:<\/strong> Consider the function f:R \u2192 R defined as $f(x) = x^{2}$.&nbsp;<\/p>\n\n\n\n<p>$y = x^{2}$<\/p>\n\n\n\n<p>$x = \\pm \\sqrt{y}$<\/p>\n\n\n\n<p>The square of any real number is non-negative.<\/p>\n\n\n\n<p>It means that $y \\ge 0$.<\/p>\n\n\n\n<p>Thus, for $y \\le 0$, we cannot find an element x such that f(x) = y.<\/p>\n\n\n\n<p>Thus, the range of f(x) is the set of non-negative real numbers and the negative real numbers are not in the image of f(x).&nbsp;<\/p>\n\n\n\n<p>As a result, f(x) is not onto.<\/p>\n\n\n\n<p><strong>Note:<\/strong> If you restrict the co-domain to \u211d<sup>+<\/sup>\u222a{0}, which is the set of non-negative real numbers, the function becomes onto.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-number-of-onto-functions-formula\">Number of Onto Functions Formula<\/h2>\n\n\n\n<p>Consider a function f:A \u2192 B such that A has n elements and B has m elements.<\/p>\n\n\n\n<p>| A | = n<\/p>\n\n\n\n<p>| B | = m<\/p>\n\n\n\n<p>Total number of onto functions = Total number of functions \u2013 Number of functions that are not onto<\/p>\n\n\n\n<p>where<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Total number of onto functions $= m^{n}$&nbsp;<\/li>\n\n\n\n<li>Number of functions that are not onto =<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"564\" height=\"44\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/number-of-functions-that-are-not-onto.png\" alt=\"Number of functions that are not onto\" class=\"wp-image-34798\" title=\"Number of functions that are not onto\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/number-of-functions-that-are-not-onto.png 564w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/10\/number-of-functions-that-are-not-onto-300x23.png 300w\" sizes=\"auto, (max-width: 564px) 100vw, 564px\" \/><\/figure>\n\n\n\n<p>If n &lt; m, the number of onto functions = 0<\/p>\n\n\n\n<p>If n = m, the number of onto functions = m!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-properties-of-onto-function\">Properties of Onto Function<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The range of an onto function is equal to its co-domain.&nbsp;<\/li>\n\n\n\n<li>Each element in the codomain has at least one preimage. In other words, the function \u201chits\u201d every element in the output set.<\/li>\n\n\n\n<li>Every onto function has a right inverse. Every function that has a right inverse is an onto function.<\/li>\n\n\n\n<li>Every onto function is not necessarily one-to-one.<\/li>\n\n\n\n<li>Compositions of onto functions are also onto. This means that if f: A \u2192 B and g: B \u2192 C are both onto, then g \u25e6 f: A \u2192 C is also onto.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-onto-function-graph\">Onto Function Graph<\/h2>\n\n\n\n<p>You can compare the range and co-domain using the graph and determine whether the function is onto or not. If the range equals codomain, the function is onto.<\/p>\n\n\n\n<p>Thus, for onto functions, the graph must reach all the y-values in the codomain. If the graph does not hit all the y-values, it is not an onto function. The graph of an <strong>onto function<\/strong> will pass through every point in the range exactly once. It means that no element in the range is left out.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-facts-about-onto-functions\">Facts about Onto Functions<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>If a function is one-one and onto, it is called a bijective function.<\/li>\n\n\n\n<li>Every finite set with at least two elements can be matched one-to-one with a proper subset of itself.<\/li>\n\n\n\n<li>If a function&#8217;s range equals its codomain, it is onto.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about onto functions, their properties, and also about how to identify onto functions with examples. Let\u2019s recall and use these concepts to solve a few examples and MCQs!&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-solved-examples-on-onto-function\">Solved Examples on Onto Function<\/h2>\n\n\n\n<p><strong>1. Show that the function f: R \u2192<\/strong> <strong>R defined as <\/strong>$f(x) = x^{3}$<strong>is onto.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>To show that f is onto, we need to show that every y-value has a corresponding x-value such that <strong>f(x) =y<\/strong>.&nbsp;<\/p>\n\n\n\n<p>Let y be any real number. Then we can solve for x by taking the cube root of both sides:<\/p>\n\n\n\n<p>y&nbsp; $= x^{3}$<\/p>\n\n\n\n<p>x $= y^{(1\/3)}<\/p>\n\n\n\n<p>Thus, for each <strong>y<\/strong><strong>, there is <\/strong>$x = y^{(1\/3)} such that <strong>f(x) = y<\/strong><strong>.<\/strong><\/p>\n\n\n\n<p>Thus, <strong>f<\/strong> is onto.<\/p>\n\n\n\n<p><strong>2. Determine whether the function f: R \u2192<\/strong> <strong>R defines as f(x) = 2x &#8211; 1 is onto.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>Let <strong>f(x) = y<\/strong> = 2x &#8211; 1&nbsp;<\/p>\n\n\n\n<p>2x = y + 1<\/p>\n\n\n\n<p>x $= \\frac{(y + 1)}{2}$&nbsp;<\/p>\n\n\n\n<p>For every <strong>y<\/strong><strong>, we have <\/strong>$x =\\frac{(y + 1)}{2}$, such that <strong>f(x) = y<\/strong><strong>.<\/strong><\/p>\n\n\n\n<p>Thus, f is onto.<\/p>\n\n\n\n<p><strong>3. Let g: R \u2192 [0, \u221e) be defined as <\/strong><strong>g(x) = x\u00b2<\/strong><strong>. Is g an onto function?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p><strong>g(x) = x\u00b2<\/strong><\/p>\n\n\n\n<p><strong>y = x\u00b2<\/strong><\/p>\n\n\n\n<p><strong>x = <\/strong>$\\sqrt{y}$<\/p>\n\n\n\n<p>Every non-negative real number has a square root.<\/p>\n\n\n\n<p>For every y in [0, \u221e), we have x=y such that <strong>g(x) = y<\/strong><strong>.<\/strong><\/p>\n\n\n\n<p>Yes, g is an <strong>onto function<\/strong> since every non-negative real number has a preimage in R.<\/p>\n\n\n\n<p><strong>4. Let f: R \u2192 R be a function defined by <\/strong><strong>f(x) = 3x + 1<\/strong><strong>. Is f onto?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>For f(x) = 3x + 1, let y be an element in the range of f.&nbsp;<\/p>\n\n\n\n<p>Then, we need to find <strong>x<\/strong> such that f(<strong>x<\/strong>) = y.<\/p>\n\n\n\n<p>y = 3x + 1<\/p>\n\n\n\n<p>3x = y &#8211; 1<\/p>\n\n\n\n<p>x $= \\frac{y\\;-\\;1}{3}$<\/p>\n\n\n\n<p>Every y in R has a preimage $x = \\frac{y\\;-\\;1}{3}$ in R.<\/p>\n\n\n\n<p>Thus, function is onto.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-practice-problems-on-onto-function\">Practice Problems on Onto Function<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Onto Function: Definition with Examples<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">A function is onto if<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">Range = domain<\/div><div class=\"spq_answer_block\" data-value=\"1\">Range = codomain<\/div><div class=\"spq_answer_block\" data-value=\"2\">Domain = Codomain<\/div><div class=\"spq_answer_block\" data-value=\"3\">None of the above<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: Range = codomain<br\/>If range = codomain, the function is onto.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">Onto function is also known as a<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">surjective function<\/div><div class=\"spq_answer_block\" data-value=\"1\">injective function<\/div><div class=\"spq_answer_block\" data-value=\"2\">bijective function<\/div><div class=\"spq_answer_block\" data-value=\"3\">many-to-one function<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: surjective function<br\/>Onto function is also known as surjective function.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">For an onto function f:$A\\rightarrow B$, every b  B has a  A such that <\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">a = b<\/div><div class=\"spq_answer_block\" data-value=\"1\">f(b) = a<\/div><div class=\"spq_answer_block\" data-value=\"2\">f(a) = b<\/div><div class=\"spq_answer_block\" data-value=\"3\">f(a) = f(b)<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: f(a) = b<br\/>For an onto function f:$A\\rightarrow B$, every b  B has a  A such that f(a) = b.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Let f: $R \\rightarrow R$. Which of the following will be an onto function?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">f(x) = x<\/div><div class=\"spq_answer_block\" data-value=\"1\">f(x) = c<\/div><div class=\"spq_answer_block\" data-value=\"2\">f(x) = |x|<\/div><div class=\"spq_answer_block\" data-value=\"3\">f(x) $= x^{2}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: f(x) = x<br\/>f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">Consider f(x) =c defined on R. It will be an onto function if the range and the codomain are equal to ______<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">{x}<\/div><div class=\"spq_answer_block\" data-value=\"1\">{1}<\/div><div class=\"spq_answer_block\" data-value=\"2\">{c}<\/div><div class=\"spq_answer_block\" data-value=\"3\">{0}<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: {1}<br\/>The range has only one element, c. 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               \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"surjective function\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Onto function is also known as surjective function.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Onto function is also known as surjective function.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that \",\n                    \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that \",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a = b\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"f(b) = a\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"f(a) = f(b)\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"f(a) = b\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"For an onto function f:$$A\\\\rightarrow B$$, every b  B has a  A such that f(a) = b.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Let f: $$R \\\\rightarrow R$$. Which of the following will be an onto function?\",\n                    \"text\": \"Let f: $$R \\\\rightarrow R$$. Which of the following will be an onto function?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"f(x) = c\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"f(x) = |x|\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"f(x) $$= x^{2}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"f(x) = x\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"f(x) =x is an identity function. Every element is mapped to itself. So, every element has a preimage. Thus, it is an onto function.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Consider f(x) =c defined on R. It will be an onto function if the range and the codomain are equal to ______\",\n                    \"text\": \"Consider f(x) =c defined on R. It will be an onto function if the range and the codomain are equal to ______\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"{x}\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"{c}\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"{0}\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"{1}\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The range has only one element, c. Thus, if range = codomain = {c}, then the function will be onto.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-frequently-asked-questions-about-the-onto-function\">Frequently Asked Questions about the Onto Function<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-b9b9d320-5636-4735-8087-614600507678\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-b9b9d320-5636-4735-8087-614600507678\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b9b9d320-5636-4735-8087-614600507678\"><strong>What is the difference between an onto function and a one-one function?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-b9b9d320-5636-4735-8087-614600507678\">\n\n<p>A one-one function maps each element in the domain to a unique element in the codomain. Thus, every element of the range corresponds to exactly one element of its domain. The output values are never repeated.<\/p>\n\n\n\n<p><strong>Onto function<\/strong> maps every element in the codomain to at least one element in the domain. The function covers or hits each element in the codomain, thereby range = codomain.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-b9b9d320-5636-4735-8087-614600507678\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b9b9d320-5636-4735-8087-614600507678\"><strong>Can a function be both one-to-one and onto?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-b9b9d320-5636-4735-8087-614600507678\">\n\n<p>Yes, a function can be both one-to-one and onto. Such a function is called a bijection.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-b9b9d320-5636-4735-8087-614600507678\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b9b9d320-5636-4735-8087-614600507678\"><strong>When is a function not onto?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-b9b9d320-5636-4735-8087-614600507678\">\n\n<p>If there is an element y in the co-domain for which there is no x value in the domain such that f(x) = y.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-b9b9d320-5636-4735-8087-614600507678\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b9b9d320-5636-4735-8087-614600507678\"><strong>What is the right inverse of a function?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-b9b9d320-5636-4735-8087-614600507678\">\n\n<p>For a function f : A\u2192B, the function g : B\u2192A is a right inverse of f if f\u2218g = I. It is called a right inverse because the inverse function is written to the right of the original function.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-b9b9d320-5636-4735-8087-614600507678\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b9b9d320-5636-4735-8087-614600507678\"><strong>What is onto vs. one-to-one function?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-b9b9d320-5636-4735-8087-614600507678\">\n\n<ul class=\"wp-block-list\">\n<li>An onto function, also known as a surjective function, maps elements from the domain to the entire codomain, ensuring each element in the codomain is covered.<\/li>\n\n\n\n<li>A one-to-one function, also called an injective function, assigns unique elements from the domain to distinct elements in the codomain, avoiding repetition.<\/li>\n<\/ul>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is an Onto Function? Onto function is also known as a surjective function. For an onto function, each element in the second set (co-domain) has a preimage.&nbsp; When we define a function from set X to set Y, the set X containing input values is called the domain of the function. The set Y &#8230; <a title=\"Onto Function: Definition, Formula, Properties, Graph, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/onto-function\" aria-label=\"More on Onto Function: Definition, Formula, Properties, Graph, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-34790","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/34790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=34790"}],"version-history":[{"count":9,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/34790\/revisions"}],"predecessor-version":[{"id":39976,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/34790\/revisions\/39976"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=34790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=34790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=34790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}