{"id":626586,"date":"2023-09-12T11:30:05","date_gmt":"2023-09-12T11:30:05","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=626586"},"modified":"2023-09-12T11:30:12","modified_gmt":"2023-09-12T11:30:12","slug":"kc-sinha-exercise-2-3-mathematics-solution-class-12-chapter-2-phalan","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-2-3-mathematics-solution-class-12-chapter-2-phalan\/","title":{"rendered":"KC Sinha: Exercise 2.3- Mathematics Solution Class 12 Chapter 2 \u092b\u0932\u0928"},"content":{"rendered":"\n

<\/span><\/p>\n\n\n\n

Question 1<\/h4>\n\n\n\n

Let S=|1,2.3|. Determine whether the functions $f: S \\rightarrow S$ defined as below have inverse. Find $f^{-1}$ if it exists.<\/p>\n\n\n\n

(i) $f=\\{(1,2),(2,1),(3,1)\\}$<\/p>\n\n\n\n

(iii) $f=\\{(1.1),(2.2),(3.3)\\}$<\/p>\n\n\n\n

Question 2<\/h4>\n\n\n\n

State with reason whether following functions have inverse. Also find the inverse if it exists.<\/p>\n\n\n\n

(i) $g:(5,6,7,8) \\rightarrow\\{1,2,3,4\\}$ with $g=\\{(5,4),(6,3),(7,4),(8.2)\\}$<\/p>\n\n\n\n

(ii) $f:|1,2,3,4| \\rightarrow\\{10\\}$ with $f=\\{(1,10),(2,10) \\cdot(3,10),(4,10)\\}$<\/p>\n\n\n\n

(iii) $h=\\{2,3,4.5\\} \\rightarrow\\{7.9 .11 .13\\}$ with $h=\\{(2.7),(3,9),(4,11),(5,13)\\}$<\/p>\n\n\n\n

Question 3<\/h4>\n\n\n\n

Let $f: R \\rightarrow R$ be defined by $f(x)=3 x+2$ Show that $f$ is invertible. Find $f^{-1}: R \\rightarrow R$.<\/p>\n\n\n\n

Question 4<\/h4>\n\n\n\n

If $f: R \\rightarrow R$ is defined by $f(x)=2 x+7$. Prove that $f$ is a bijection. Also. find the inverse of $f$.<\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

Question 5<\/h4>\n\n\n\n

Let R be the set of all real numbers. Show that the function $f: R \\rightarrow R$ defined by $f(x)=\\frac{3 x+1}{2}$ for all $x \\in R$ is bijective. Also find the inverse of $f$.<\/p>\n\n\n\n

Question 6<\/h4>\n\n\n\n

Consider the function $f: R^{+} \\rightarrow R^{+}$defined by $f(x)=e^{x} . \\forall x \\in R^{+}$. where $R^{+}$ denotes the set of all +ve real numbers. Find inverse function of $f$ if it exists. <\/p>\n\n\n\n

Question 7<\/h4>\n\n\n\n

If $f:[0, \\infty) \\rightarrow[2, \\infty)$ be defined by $f(x)=x^{2}+2, \\forall x \\in R$. Then find $f^{-1}$.<\/p>\n\n\n\n

Question 8<\/h4>\n\n\n\n

If $A=\\{1,2,3,4\\}, B=\\{2,4,6,8\\}$ and $f: A \\rightarrow B$ is given by $f(x)=2 x$, then write $f$ and $f^{-1}$ as a set of ordered pairs.<\/p>\n\n\n\n

Question 9<\/h4>\n\n\n\n

Let $f: R \\rightarrow R$ given by $f(x)=4 x+3$. Show that $f$ is invertible. Find the inverse of $f$.<\/p>\n\n\n\n

Question 10<\/h4>\n\n\n\n

Let $f: N \\rightarrow Y$ be a function defined as $f(x)=4 x+3$, where $Y=\\{y \\in N: y=4 x+3$ for some $x \\in N\\}$. Show that $f$ is invertible. Find the inverse.<\/p>\n\n\n\n

Question 11<\/h4>\n\n\n\n

Let $f: R_{+} \\rightarrow(4, \\infty)$ be given by $f(x)=x^{2}+4$. Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y)=\\sqrt{y-4}$.<\/p>\n\n\n\n

Question 12<\/h4>\n\n\n\n

Let $f: N \\rightarrow R$ be a function defined as $f(x)=4 x^{2}+12 x+15$. Show that $f: N \\rightarrow$ range $f$ is invertible. Find the inverse of $f$.<\/p>\n\n\n\n

Question 13<\/h4>\n\n\n\n

Let $Y=\\left\\{n^{2}: n \\in N\\right\\} \\subset N$, consider $f: N \\rightarrow Y$ as $f(n)=n^{2}$. Show that $f$ is invertible. Find the inverse of $f$.<\/p>\n\n\n\n

Question 14<\/h4>\n\n\n\n

Consider $f:\\{1,2,3\\} \\rightarrow\\{a, b, c\\}$ given by $f(1)=a, f(2)=b$ and $f(3)=c$. Find the inverse $\\left(f^{-1}\\right)^{-1}$ of $f^{-1}$. Also show that $\\left(f^{-1}\\right)^{-1}=f$.<\/p>\n\n\n\n

Question 15<\/h4>\n\n\n\n

Let $F: R \\rightarrow R$ be defined as $f(x)=10 x+7$.<\/p>\n\n\n\n

Find the function $g: R \\rightarrow R$ such that gof $=f o g=I_{R}$<\/p>\n\n\n\n

Question 16<\/h4>\n\n\n\n

If the function $f: R \\rightarrow R$ be defined by $f(x)=2 x-3$ and $g: R \\rightarrow R$ by $g(x)=x^{3}+5$, then find the value of $(f o g)^{-1} x$<\/p>\n\n\n\n

\n
KC Sinha Solutions<\/a><\/div>\n\n\n\n
KC Sinha Class 12 Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Question 1 Let S=|1,2.3|. Determine whether the functions $f: S \\rightarrow S$ defined as below have inverse. Find $f^{-1}$ if it exists. (i) $f=\\{(1,2),(2,1),(3,1)\\}$ (iii) $f=\\{(1.1),(2.2),(3.3)\\}$ Question 2 State with reason whether following functions have inverse. Also find the inverse if it exists. (i) $g:(5,6,7,8) \\rightarrow\\{1,2,3,4\\}$ with $g=\\{(5,4),(6,3),(7,4),(8.2)\\}$ (ii) $f:|1,2,3,4| \\rightarrow\\{10\\}$ with $f=\\{(1,10),(2,10) \\cdot(3,10),(4,10)\\}$ (iii) […]<\/p>\n","protected":false},"author":302,"featured_media":626589,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[25],"tags":[],"boards":[],"class_list":["post-626586","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","entry"],"yoast_head":"\nKC Sinha: Exercise 2.3- Mathematics Solution Class 12 Chapter 2 \u092b\u0932\u0928 - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"Question 1 Let S=|1,2.3|. 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