inverse of f(x)=(x=2)/(5x-1)

Question

inverse of the function x  for x plus two  divided by five x minus 1

Answer 1

Let y=f(x)=(x+2)/(5x-1),

y(5x-1)=x+2,

5xy-y=x+2,

5xy-x=y+2,

x(5y-1)=y+2,

x=(y+2)/(5y-1).

Let x=g(y)=(y+2)/(5y-1), so g(x)=(x+2)/(5x-1)=f(x).

But g(x)=f^-1(x) so f^-1(x)=f(x).

To check this, remember that f(f^-1(x))=x by definition of the inverse.

f(f^-1(x))=(f^-1(x)+2)/(5f^-1(x)-1)=((x+2)/(5x-1)+2)/(5(x+2)/(5x-1)-1).

Multiply top and bottom by 5x-1:

(x+2+10x-2)/(5x+10-5x+1)=11x/11=x,

which proves that the inverse of f(x)=(x+2)/(5x-1) is the same as f(x).

Also, f^-1(f(x))=x since the function and its inverse are identical.