Ex 1.3 , 6 - Chapter 1 Class 12 Relation and Functions (Term 1)
Last updated at Jan. 28, 2020 by
Last updated at Jan. 28, 2020 by
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Ex 1.3, 6 Show that f: [β1, 1] β R, given by f(x) = π₯/(π₯ + 2) is one-one. Find the inverse of the function f: [β1, 1] β Range f. (Hint: For y β Range f, y = f(x) = π₯/(π₯ + 2) , for some x in [β1, 1], i.e., x = 2π¦/(1 β π¦) ) f(x) = x/(x+2) Check one-one f(x1) = π₯1/(π₯1 + 2) f(x2) = π₯2/(π₯2 + 2) Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f(x1) = f(x2) π₯1/(π₯1 + 2) = π₯2/(π₯2 + 2) x1(x2 + 2) = x2(x1 + 2) x1x2 + 2x1 = x2x1 + 2x2 x1x2 β x2x1 + 2x1 = 2x2 0 + 2x1 = 2x2 2x1 = 2x2 x1 = x2 Hence, if f(x1) = f(x2) , then x1 = x2 β΄ f is one-one Checking onto f(x) = π₯/(π₯ + 2) Putting f(x) = y y = π₯/(π₯ + 2) y(x + 2) = x yx + 2y = x yx β x = β2y x(y β 1) = β2y x = (β2π¦ )/(π¦ β1) x = (β2π¦ )/(β1(βπ¦ + 1) ) x = (2π¦ )/((1 β π¦) ) Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f((2π¦ )/((1 β π¦) )) = ((2π¦ )/((1 β π¦) ))/((2π¦ )/((1 β π¦) ) + 2) = ((2π¦ )/((1 β π¦) ))/((2π¦ + 2(1 β π¦) )/((1 β π¦) )) = 2π¦/(2π¦ + 2 β 2π¦) = y Thus, for every y β Range f, there exists x β [β1, 1] such that f(x) = y Hence, f is onto Since f(x) is one-one and onto, So, f(x) is invertible And Inverse of x = π^(β1) (π¦) = (2π¦ )/((1 β π¦) ) , y β 1 Note: Here, y β Range f is important Inverse is not defined for y β R Because denominator in (2π¦ )/((1 β π¦) ) will be 0 if y = 1
Finding Inverse
Inverse of a function
How to check if function has inverse?
Example 22 Deleted for CBSE Board 2022 Exams
Ex 1.3, 5 (i) Deleted for CBSE Board 2022 Exams
How to find Inverse?
Example 28 (a) Deleted for CBSE Board 2022 Exams
Misc 11 (i) Important Deleted for CBSE Board 2022 Exams
Ex 1.3, 11 Deleted for CBSE Board 2022 Exams
Example 27 Important Deleted for CBSE Board 2022 Exams
Misc 1 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 6 Deleted for CBSE Board 2022 Exams You are here
Ex 1.3, 14 (MCQ) Important Deleted for CBSE Board 2022 Exams
Example 23 Important Deleted for CBSE Board 2022 Exams
Misc 2 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 4 Deleted for CBSE Board 2022 Exams
Example 24 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 8 Important Deleted for CBSE Board 2022 Exams
Example 25 Important Deleted for CBSE Board 2022 Exams
Ex 1.3 , 9 Important Deleted for CBSE Board 2022 Exams
Finding Inverse
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