Show that the function f in A= R-{2/3} defined as f(x)=4x+3/6x-4 is one -one and onto Hence find the - Maths - Relations and Functions

Question

Show that the function f in A= R-{2/3} defined as f(x)=4x+3/6x-4 is
one -one and onto Hence find the inverse of f

Rishabh Mittal · Accepted Answer

Solution :

fx=4x+36x-4For One-One :Let x1, x2 ∈A
 Then    fx1 = fx2⇒4x1+36x1-4=4x2+36x2-4⇒4x1+36x2-4=4x2+36x1-4⇒24x1x2-16x1+18x2-12=24x1x2-16x2+18x1-12⇒-16x1+18x2=-16x2+18x1⇒-34x1=-34x2⇒x1=xso, fx is one-oneFor onto :Let y=fx = 4x+36x-4⇒y6x-4=4x+3⇒6xy - 4y =4x + 3⇒6xy -4x =3 +4y⇒x6y-4=3+4y⇒x=3+4y6y-4Now , f3+4y6y-4=43+4y6y-4+363+4y6y-4-4                             =12+16y+18y-1218+24y-24y+16=34y34=ySo, fx is Onto
and f-1x=3+4x6x-4

Show that the function f in A= R-{2/3} defined as f(x)=4x+3/6x-4 is one -one and onto. Hence find the inverse of f.