# Misc. 4 - Chapter 1 Class 12 Relation and Functions (Term 1)

Last updated at Jan. 28, 2020 by Teachoo

Last updated at Jan. 28, 2020 by Teachoo

Transcript

Misc 4 Show that function f: R โ {x โ R: โ1 < x < 1} defined by f(x) = x/(1 + |๐ฅ| ) , x โ R is one-one and onto function. f: R โ {x โ R: โ1 < x < 1} f(x) = x/(1 + |๐ฅ| ) We know that |๐ฅ| = {โ( ๐ฅ , ๐ฅโฅ0 @โ๐ฅ , ๐ฅ<0)โค So, ๐(๐ฅ)={โ(๐ฅ/(1 + ๐ฅ), ๐ฅโฅ0@&๐ฅ/(1 โ ๐ฅ), ๐ฅ<0)โค For x โฅ 0 f(x1) = ๐ฅ_1/(1 + ๐ฅ_1 ) f(x2) = ๐ฅ_2/(1 + ๐ฅ_2 ) Putting f(x1) = f(x2) ๐ฅ_1/(1 + ๐ฅ_1 ) = ๐ฅ_2/(1 + ๐ฅ_2 ) ๐ฅ_1 (1 + ๐ฅ_2)= ๐ฅ_2 (1 + ๐ฅ_1) ๐ฅ_1+๐ฅ_1 ๐ฅ_2= ๐ฅ_2 +๐ฅ_2 ๐ฅ_1 ๐ฅ_1= ๐ฅ_2 For x < 0 f(x1) = ๐ฅ_1/(1 โ ๐ฅ_1 ) f(x2) = ๐ฅ_2/(1 โ ๐ฅ_2 ) Putting f(x1) = f(x2) ๐ฅ_1/(1 โ ๐ฅ_1 ) = ๐ฅ_2/(1 โ ๐ฅ_2 ) ๐ฅ_1 (1 โ ๐ฅ_2)= ๐ฅ_2 (1 โ ๐ฅ_1) ๐ฅ_1โ๐ฅ_1 ๐ฅ_2= ๐ฅ_2 โ๐ฅ_2 ๐ฅ_1 ๐ฅ_1= ๐ฅ_2 Hence, if f(x1) = f(x2) , then x1 = x2 โด f is one-one Checking onto ๐(๐ฅ)={โ(๐ฅ/(1 + ๐ฅ), ๐ฅโฅ0@&๐ฅ/(1 โ ๐ฅ), ๐ฅ<0)โค For x โฅ 0 f(x) = ๐ฅ/(1 + ๐ฅ) Let f(x) = y, "y = " ๐ฅ/(1 + ๐ฅ) y(1 + x) = x y + xy = x y = x โ xy x โ xy = y x(1 โ y) = y x = ๐ฆ/(1 โ ๐ฆ) For x < 0 f(x) = ๐ฅ/(1 โ ๐ฅ) Let f(x) = y "y = " ๐ฅ/(1 โ ๐ฅ) y(1 โ x) = x y โ xy = x y = x + xy x + xy = y x(1 + y) = y x = ๐ฆ/(1 + ๐ฆ) Thus, x = ๐ฆ/(1 โ ๐ฆ) , for x โฅ 0 & x = ๐ฆ/(1 + ๐ฆ) , for x < 0 Here, y โ {x โ R: โ1 < x < 1} i.e. Value of y is from โ1 to 1 , โ 1 < y < 1 If y = 1, x = ๐ฆ/(1 โ ๐ฆ) will be not defined, If y = โ1, x = ๐ฆ/(1 + ๐ฆ) will be not defined, But here โ 1 < y < 1 So, x is defined for all values of y. & x โ R โด f is onto Hence, f is one-one and onto.

Miscellaneous

Misc 1
Deleted for CBSE Board 2022 Exams

Misc 2 Deleted for CBSE Board 2022 Exams

Misc 3 Important Deleted for CBSE Board 2022 Exams

Misc. 4 Important You are here

Misc 5

Misc 6 Deleted for CBSE Board 2022 Exams

Misc 7 Deleted for CBSE Board 2022 Exams

Misc. 8 Important

Misc 9 Important Deleted for CBSE Board 2022 Exams

Misc 10 Important

Misc 11 (i) Important Deleted for CBSE Board 2022 Exams

Misc 11 (ii) Deleted for CBSE Board 2022 Exams

Misc 12 Deleted for CBSE Board 2022 Exams

Misc 13 Important Deleted for CBSE Board 2022 Exams

Misc 14 Important Deleted for CBSE Board 2022 Exams

Misc 15

Misc 16 (MCQ) Important

Misc 17 (MCQ) Important

Misc 18 Deleted for CBSE Board 2022 Exams

Misc 19 (MCQ) Important Deleted for CBSE Board 2022 Exams

Chapter 1 Class 12 Relation and Functions (Term 1)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.