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Misc 4 - Show f(x) = x / 1 + |x| is one-one onto - Miscellaneous

Misc. 4 - Chapter 1 Class 12 Relation and Functions - Part 2
Misc. 4 - Chapter 1 Class 12 Relation and Functions - Part 3
Misc. 4 - Chapter 1 Class 12 Relation and Functions - Part 4
Misc. 4 - Chapter 1 Class 12 Relation and Functions - Part 5
Misc. 4 - Chapter 1 Class 12 Relation and Functions - Part 6

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Misc 4 Show that function f: R β†’ {x ∈ R: βˆ’1 < x < 1} defined by f(x) = x/(1 + |π‘₯| ) , x ∈ R is one-one and onto function. f: R β†’ {x ∈ R: βˆ’1 < x < 1} f(x) = x/(1 + |π‘₯| ) We know that |π‘₯| = {β–ˆ( π‘₯ , π‘₯β‰₯0 @βˆ’π‘₯ , π‘₯<0)─ So, 𝑓(π‘₯)={β–ˆ(π‘₯/(1 + π‘₯), π‘₯β‰₯0@&π‘₯/(1 βˆ’ π‘₯), π‘₯<0)─ For x β‰₯ 0 f(x1) = π‘₯_1/(1 + π‘₯_1 ) f(x2) = π‘₯_2/(1 + π‘₯_2 ) Putting f(x1) = f(x2) π‘₯_1/(1 + π‘₯_1 ) = π‘₯_2/(1 + π‘₯_2 ) π‘₯_1 (1 + π‘₯_2)= π‘₯_2 (1 + π‘₯_1) π‘₯_1+π‘₯_1 π‘₯_2= π‘₯_2 +π‘₯_2 π‘₯_1 π‘₯_1= π‘₯_2 For x < 0 f(x1) = π‘₯_1/(1 βˆ’ π‘₯_1 ) f(x2) = π‘₯_2/(1 βˆ’ π‘₯_2 ) Putting f(x1) = f(x2) π‘₯_1/(1 βˆ’ π‘₯_1 ) = π‘₯_2/(1 βˆ’ π‘₯_2 ) π‘₯_1 (1 βˆ’ π‘₯_2)= π‘₯_2 (1 βˆ’ π‘₯_1) π‘₯_1βˆ’π‘₯_1 π‘₯_2= π‘₯_2 βˆ’π‘₯_2 π‘₯_1 π‘₯_1= π‘₯_2 Hence, if f(x1) = f(x2) , then x1 = x2 ∴ f is one-one Checking onto 𝑓(π‘₯)={β–ˆ(π‘₯/(1 + π‘₯), π‘₯β‰₯0@&π‘₯/(1 βˆ’ π‘₯), π‘₯<0)─ For x β‰₯ 0 f(x) = π‘₯/(1 + π‘₯) Let f(x) = y, "y = " π‘₯/(1 + π‘₯) y(1 + x) = x y + xy = x y = x – xy x – xy = y x(1 – y) = y x = 𝑦/(1 βˆ’ 𝑦) For x < 0 f(x) = π‘₯/(1 βˆ’ π‘₯) Let f(x) = y "y = " π‘₯/(1 βˆ’ π‘₯) y(1 – x) = x y – xy = x y = x + xy x + xy = y x(1 + y) = y x = 𝑦/(1 + 𝑦) Thus, x = 𝑦/(1 βˆ’ 𝑦) , for x β‰₯ 0 & x = 𝑦/(1 + 𝑦) , for x < 0 Here, y ∈ {x ∈ R: βˆ’1 < x < 1} i.e. Value of y is from –1 to 1 , – 1 < y < 1 If y = 1, x = 𝑦/(1 βˆ’ 𝑦) will be not defined, If y = –1, x = 𝑦/(1 + 𝑦) will be not defined, But here – 1 < y < 1 So, x is defined for all values of y. & x ∈ R ∴ f is onto Hence, f is one-one and onto.

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.