Let f: A → B be a function defined as f(x) = (2x + 3)/(x - 3) , where
A = R − {3} and B = R − {2}. Is the function f one–one and onto? Is f invertible? If yes, then find its inverse





CBSE Class 12 Sample Paper for 2020 Boards
CBSE Class 12 Sample Paper for 2020 Boards
Last updated at Dec. 16, 2024 by Teachoo
Transcript
Question 27 Let π: A β B be a function defined as π(π₯)=(2π₯ + 3)/(π₯ β 3) , where A = R β {3} and B = R β {2}. Is the function f oneβone and onto? Is f invertible? If yes, then find its inverse π(π₯)=(2π₯ + 3)/(π₯ β 3) Checking one-one Let π₯_1 , π₯_2 β A π(π₯_1 )=(2π₯_1+ 3)/(π₯_1β 3) π(π₯_2 )=(2π₯_2+ 3)/(π₯_2 β 3) One-one Steps 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting π(π₯_1 ) = π(π₯_2 ) (2π₯_1+ 3)/(π₯_1β 3) = (2π₯_2+ 3)/(π₯_2 β 3) (2π₯_1+ 3) (π₯_2 β 3) = (2π₯_2+ 3) (π₯_1 β 3) 2π₯_1 "(" π₯_2 β 3")"+ 3 (π₯_2 β 3) = 2π₯_2 "(" π₯_1 β 3")"+ 3(π₯_1 β 3) 2π₯_1 π₯_2 β6π₯_1+ 3π₯_2β9 = 2π₯_2 π₯_1 β6π₯_2+ 3π₯_1 β9 β6π₯_1+ 3π₯_2 = β6π₯_2+ 3π₯_1 3π₯_2+6π₯_2 = 3π₯_1+6π₯_1 9π₯_2= 9π₯_1 π₯_2= π₯_1 If π(π₯_1 ) = π(π₯_2 ) , then π₯_1= π₯_2 β΄ f is one-one Checking onto π(π₯)=(2π₯ + 3)/(π₯ β 3) Let y β B where y = (2π₯ + 3)/(π₯ β 3) y(x β 3) = 2x + 3 xy β 3y = 2x + 3 xy β 2x = 3y + 3 x(y β 2) = 3(y + 1) x = (3(π¦ + 1))/((π¦ β 2)) For y = 2 , x is not defined But it is given that y β R β {2} Hence , x = (3(π¦ + 1))/((π¦ β 2)) β R β {3} Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f((3(π¦ + 1))/((π¦ β 2))) = (2(3(π¦ + 1)/((π¦ β 2) ))+ 3)/((3(π¦ + 1)/((π¦ β 2) ))β 3) = (((6(π¦ + 1)+3(π¦β2))/((π¦ β 2) )))/((3(π¦ + 1)/((π¦ β 2) ))β 3) Also, y = f(x) Putting y in f(x) Hence, f is onto = (2(3(π¦ + 1)/((π¦ β 2) ))+ 3)/((3(π¦ + 1)/((π¦ β 2) ))β 3) = (((6(π¦ + 1) + 3(π¦ β 2))/((π¦ β 2) )))/(((3(π¦ + 1) β 3(π¦ β 2))/((π¦ β 2) )) ) = (6(π¦ + 1) + 3(π¦ β 2))/(3(π¦ + 1) β 3(π¦ β 2)) = (6π¦ + 6 + 3π¦ β 6)/(3π¦ + 3 β 3π¦ + 6) = 9π¦/9 = y Thus, for every y β B, there exists x β A 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¦) = (3(π¦ + 1))/((π¦ β 2))