{"id":1820,"date":"2026-05-21T07:29:09","date_gmt":"2026-05-21T07:29:09","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1820"},"modified":"2026-05-21T07:29:12","modified_gmt":"2026-05-21T07:29:12","slug":"relations-and-functions-exercise-1-2-solutions-class-12","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/relations-and-functions-exercise-1-2-solutions-class-12\/","title":{"rendered":"Relations and Functions Exercise 1.2 Solutions Class 12"},"content":{"rendered":"<div class=\"mymoc-top mymoc-entity-placement\" id=\"mymoc-1970052182\"><div id=\"mymoc-4119728962\"><a href=\"https:\/\/amzn.to\/3S4CntS\" aria-label=\"Stationeries\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries.png 1303w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-300x89.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-1024x304.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/Stationeries-768x228.png 768w\" sizes=\"(max-width: 1303px) 100vw, 1303px\" width=\"1080\" height=\"100\"><\/a><\/div><\/div><div class=\"pdfprnt-buttons pdfprnt-buttons-post pdfprnt-top-bottom-right\"><a href=\"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1820?print=print\" class=\"pdfprnt-button pdfprnt-button-print\" target=\"_blank\"><img decoding=\"async\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/plugins\/pdf-print\/images\/print.png\" alt=\"image_print\" title=\"Print Content\"><span class=\"pdfprnt-button-title pdfprnt-button-print-title\">Print<\/span><\/a> <span class=\"pdfprnt-count-generation\">8<\/span><\/div>\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Short Intro<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 1 Exercise 1.2 &ndash; Relations and Functions based on the latest NCERT syllabus. This exercise covers one-one functions, onto functions, bijective functions, injective and surjective mappings, modulus function, signum function, and greatest integer function in an easy and exam-oriented way.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Item<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Board<\/td><td>NCERT \/ CBSE<\/td><\/tr><tr><td>Class<\/td><td>12<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Chapter<\/td><td>Relations and Functions<\/td><\/tr><tr><td>Exercise<\/td><td>1.2<\/td><\/tr><tr><td>Main Topics<\/td><td>One-One &amp; Onto Functions<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>One-One (Injective) Function<\/li>\n\n\n\n<li>Onto (Surjective) Function<\/li>\n\n\n\n<li>Bijective Function<\/li>\n\n\n\n<li>Many-One Function<\/li>\n\n\n\n<li>Greatest Integer Function<\/li>\n\n\n\n<li>Modulus Function<\/li>\n\n\n\n<li>Signum Function<\/li>\n\n\n\n<li>Domain, Co-domain &amp; Range<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter explains different types of functions and mappings between sets.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">One-One Function<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x&#8321;) = f(x&#8322;) &rArr; x&#8321; = x&#8322;<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Onto Function<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>For every y &isin; Y, there exists x &isin; X such that f(x) = y<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Bijective Function<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>A function which is both one-one and onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Modulus Function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mi mathvariant=\"normal\">&#8739;<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=|x|<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"552\" height=\"339\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-19.png\" alt=\"image\" class=\"wp-image-1838\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-19.png 552w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-19-300x184.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Greatest Integer Function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=[x]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"555\" height=\"349\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-18.png\" alt=\"image\" class=\"wp-image-1837\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-18.png 555w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-18-300x189.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\"><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Questions &amp; Step-by-step Solutions<\/h1>\n\n\n\n<h1 class=\"wp-block-heading\">Question 1<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Show that the function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mi>x<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=\\frac{1}{x}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"552\" height=\"348\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-17.png\" alt=\"image\" class=\"wp-image-1836\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-17.png 552w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-17-300x189.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">is one-one and onto on R*.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/relations-and-functions-exercise-1-1-solutions-class-12\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Relations and Functions Exercise 1.1 Solutions Class 12<\/span><\/a><\/div><p class=\"wp-block-paragraph\">Let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x&#8321;) = f(x&#8322;)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>1\/x&#8321; = 1\/x&#8322;<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x&#8321; = x&#8322;<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence, the function is one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now for any:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>y &isin; R*<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">choose:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x = 1\/y<\/code><\/pre>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/miscellaneous-exercise-solutions-class-12-relations-functions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Miscellaneous Exercise Solutions Class 12 Relations Functions<\/span><\/a><\/div><p class=\"wp-block-paragraph\">Then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x) = y<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore, the function is onto.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p><div class=\"mymoc-middle mymoc-entity-placement\" id=\"mymoc-1734343063\"><div id=\"mymoc-1633559396\"><a href=\"https:\/\/amzn.to\/4uMTpup\" aria-label=\"books\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/books.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/books.png 1301w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/books-300x56.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/books-1024x190.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/books-768x143.png 768w\" sizes=\"(max-width: 1301px) 100vw, 1301px\" width=\"1301\" height=\"242\"><\/a><\/div><\/div>\n\n\n\n<pre class=\"wp-block-code\"><code>f is bijective.<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">The result is not true if domain is N because negative real numbers cannot be images.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 2<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Check injectivity and surjectivity.<\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(i)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"556\" height=\"337\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-16.png\" alt=\"image\" class=\"wp-image-1835\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-16.png 556w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-16-300x182.png 300w\" sizes=\"(max-width: 556px) 100vw, 556px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">for:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f : N &rarr; N<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<div class=\"internal-linking-related-contents\"><a href=\"https:\/\/mymockmate.com\/notes\/inverse-trigonometric-functions-miscellaneous-solutions\/\" class=\"template-2\"><span class=\"cta\">Related Topic to Read more<\/span><span class=\"postTitle\">Inverse Trigonometric Functions Miscellaneous Solutions<\/span><\/a><\/div><p class=\"wp-block-paragraph\">If:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x&#8321;&sup2; = x&#8322;&sup2;<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">then:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x&#8321; = x&#8322;<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is not image of any natural number.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>One-one but not onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(ii)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"552\" height=\"349\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-15.png\" alt=\"image\" class=\"wp-image-1834\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-15.png 552w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-15-300x190.png 300w\" sizes=\"(max-width: 552px) 100vw, 552px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">for:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f : Z &rarr; Z<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(2) = f(&minus;2)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">the function is not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Negative integers are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Neither one-one nor onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(iii)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"555\" height=\"333\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-14.png\" alt=\"image\" class=\"wp-image-1833\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-14.png 555w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-14-300x180.png 300w\" sizes=\"(max-width: 555px) 100vw, 555px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">for:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f : R &rarr; R<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(1) = f(&minus;1)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Negative real numbers are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Neither one-one nor onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(iv)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"466\" height=\"343\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-13.png\" alt=\"image\" class=\"wp-image-1832\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-13.png 466w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-13-300x221.png 300w\" sizes=\"(max-width: 466px) 100vw, 466px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">for:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f : N &rarr; N<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Cube function is strictly increasing.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But numbers like:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">are not perfect cubes.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Not onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(v)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^3<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"493\" height=\"345\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-12.png\" alt=\"image\" class=\"wp-image-1831\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-12.png 493w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-12-300x210.png 300w\" sizes=\"(max-width: 493px) 100vw, 493px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">for:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f : Z &rarr; Z<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Cube function is one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">But integers like:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">are not cubes.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>One-one but not onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 3<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Greatest Integer Function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=[x]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"520\" height=\"336\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-11.png\" alt=\"image\" class=\"wp-image-1830\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-11.png 520w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-11-300x194.png 300w\" sizes=\"(max-width: 520px) 100vw, 520px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>[1.2] = [1.8] = 1<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">the function is not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also numbers like:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>1.5<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Neither one-one nor onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 4<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Modulus Function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi mathvariant=\"normal\">&#8739;<\/mi><mi>x<\/mi><mi mathvariant=\"normal\">&#8739;<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=|x|<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"540\" height=\"346\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-10.png\" alt=\"image\" class=\"wp-image-1829\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-10.png 540w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-10-300x192.png 300w\" sizes=\"(max-width: 540px) 100vw, 540px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>|1| = |&minus;1|<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Negative numbers are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Neither one-one nor onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 5<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Signum Function<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mn>1<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>x<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mn>0<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>&minus;<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=\\begin{cases}1,&amp;x&gt;0\\\\0,&amp;x=0\\\\-1,&amp;x&lt;0\\end{cases}<\/annotation><\/semantics><\/math>&#8203;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Many numbers have same image.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Range is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>{&minus;1,0,1}<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence not onto on R.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 6<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Function from A = {1,2,3} to B = {4,5,6,7}<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Given:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f = {(1,4),(2,5),(3,6)}<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Different elements have different images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f is one-one.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 7<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">State whether functions are one-one or onto.<\/h2>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(i)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><mo>&minus;<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=3-4x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"435\" height=\"355\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-9.png\" alt=\"image\" class=\"wp-image-1828\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-9.png 435w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-9-300x245.png 300w\" sizes=\"(max-width: 435px) 100vw, 435px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Linear function with non-zero slope.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>One-one and onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">(ii)<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=1+x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"508\" height=\"345\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-8.png\" alt=\"image\" class=\"wp-image-1827\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-8.png 508w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-8-300x204.png 300w\" sizes=\"(max-width: 508px) 100vw, 508px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(1)=f(&minus;1)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also values less than 1 are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Neither one-one nor onto.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 8<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Show that<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo separator=\"true\">,<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo separator=\"true\">,<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(a,b)=(b,a)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is bijective.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Different ordered pairs produce different images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Every pair in:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B &times; A<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">has pre-image.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hence onto.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f is bijective.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 9<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Function on natural numbers<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The function maps odd and even numbers uniquely.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Every natural number has pre-image.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>The function is bijective.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 10<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Check whether function is one-one and onto.<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>x<\/mi><mo>&minus;<\/mo><mn>2<\/mn><\/mrow><mrow><mi>x<\/mi><mo>&minus;<\/mo><mn>3<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=\\frac{x-2}{x-3}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"559\" height=\"343\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-7.png\" alt=\"image\" class=\"wp-image-1826\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-7.png 559w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-7-300x184.png 300w\" sizes=\"(max-width: 559px) 100vw, 559px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Let:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(x&#8321;)=f(x&#8322;)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">After simplification:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>x&#8321;=x&#8322;<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Also every element in:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B = R &minus; {1}<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">has pre-image.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f is onto.<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence bijective.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 11<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Choose correct answer for<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>4<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^4<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"543\" height=\"340\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-4.png\" alt=\"image\" class=\"wp-image-1823\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-4.png 543w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-4-300x188.png 300w\" sizes=\"(max-width: 543px) 100vw, 543px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Since:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>f(1)=f(&minus;1)<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">not one-one.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Negative values are not images.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Correct option:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(D) Neither one-one nor onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Question 12<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Choose correct answer for<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=3x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"463\" height=\"345\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-5.png\" alt=\"image\" class=\"wp-image-1824\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-5.png 463w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-5-300x224.png 300w\" sizes=\"(max-width: 463px) 100vw, 463px\"><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Linear function with non-zero slope.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Correct option:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(A) One-one and onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Confusing onto with one-one<\/li>\n\n\n\n<li>Forgetting co-domain<\/li>\n\n\n\n<li>Ignoring repeated outputs<\/li>\n\n\n\n<li>Wrongly assuming every linear function is onto for all domains<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Always check injective and surjective separately.<\/li>\n\n\n\n<li>Use counterexamples carefully.<\/li>\n\n\n\n<li>Remember range while checking onto.<\/li>\n\n\n\n<li>Practice graph interpretation.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function which is both one-one and onto is called:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Constant<br>B. Bijective<br>C. Many-one<br>D. Identity<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B. Bijective<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Function:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">f(x)=x^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"562\" height=\"346\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-6.png\" alt=\"image\" class=\"wp-image-1825\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-6.png 562w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/image-6-300x185.png 300w\" sizes=\"(max-width: 562px) 100vw, 562px\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">on R is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. One-one<br>B. Onto<br>C. Bijective<br>D. Neither one-one nor onto<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>D. Neither one-one nor onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The modulus function is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. Onto<br>B. One-one<br>C. Bijective<br>D. Neither one-one nor onto<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>D. Neither one-one nor onto<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h1 class=\"wp-block-heading\">FAQ Section<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">What is an injective function?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function in which different inputs have different outputs.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">What is a surjective function?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function where every element of co-domain has a pre-image.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">What is a bijective function?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A function which is both injective and surjective.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<h2 class=\"wp-block-heading\">Why is modulus function not one-one?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Because:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>|1| = |&minus;1|<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\">\n\n\n\n<p class=\"wp-block-paragraph\">&#128216; Prepare Smarter with MyMockMate!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">&#9989; Chapter-wise NCERT Solutions<br>&#9989; Important MCQs &amp; Notes<br>&#9989; Online Mock Tests<br>&#9989; Instant Result &amp; Analysis<br>&#9989; CBSE Board Preparation Support<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Start learning now on <a target=\"_blank\" rel=\"noreferrer noopener\" href=\"https:\/\/www.mymockmate.com?utm_source=chatgpt.com\">MyMockMate<\/a><\/p>\n<div class=\"pdfprnt-buttons pdfprnt-buttons-post pdfprnt-top-bottom-right\"><a href=\"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1820?print=print\" class=\"pdfprnt-button pdfprnt-button-print\" target=\"_blank\"><img decoding=\"async\" src=\"https:\/\/mymockmate.com\/notes\/wp-content\/plugins\/pdf-print\/images\/print.png\" alt=\"image_print\" title=\"Print Content\"><span class=\"pdfprnt-button-title pdfprnt-button-print-title\">Print<\/span><\/a> <span class=\"pdfprnt-count-generation\">8<\/span><\/div><div class=\"mymoc-bottom mymoc-entity-placement\" id=\"mymoc-528350500\"><div id=\"mymoc-1752208538\"><a href=\"https:\/\/amzn.to\/4fDWGbw\" aria-label=\"smartwatch\"><img src=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch.png\" alt=\"\" srcset=\"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch.png 1717w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-300x71.png 300w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-1024x243.png 1024w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-768x182.png 768w, https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/06\/smartwatch-1536x364.png 1536w\" sizes=\"(max-width: 1717px) 100vw, 1717px\" width=\"1717\" height=\"407\"><\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p> 8 Short Intro In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 1 Exercise 1.2 &ndash; Relations and Functions&#8230;<\/p>\n","protected":false},"author":1,"featured_media":1839,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"footnotes":""},"categories":[7,8,1],"tags":[196,187,175,174,186,205,202,156,46,200,197,204,199,53,47,194,195,173,176,189,188,184,201,198,203],"class_list":["post-1820","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","category-maths","category-uncategorized","tag-bijective-function","tag-cbse-board-maths-preparation","tag-cbse-class-12-maths","tag-class-12-maths","tag-class-12-maths-chapter-1-solutions","tag-class-12-maths-notes","tag-domain-and-range","tag-exercise-1-2-ncert-solutions","tag-exercise-1-2-solutions","tag-greatest-integer-function","tag-injective-function","tag-invertible-functions","tag-modulus-function","tag-ncert-maths-solutions","tag-ncert-solutions","tag-one-one-function","tag-onto-function","tag-relations-and-functions","tag-relations-and-functions-chapter-1","tag-relations-and-functions-important-questions","tag-relations-and-functions-mcqs","tag-relations-and-functions-notes","tag-signum-function","tag-surjective-function","tag-types-of-functions"],"featured_image_src":"https:\/\/mymockmate.com\/notes\/wp-content\/uploads\/2026\/05\/Relations-and-Functions-Exercise-1.2-Solutions-Class-12.png","author_info":{"display_name":"Team Mymockmate","author_link":"https:\/\/mymockmate.com\/notes\/author\/bsm_adm\/"},"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1820"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1820\/revisions"}],"predecessor-version":[{"id":1840,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1820\/revisions\/1840"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1839"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1820"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1820"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}