Find the inverse of fx 3x 12x 5 Find domain and range of f

Find the inverse of f(x) = 3x + 1/2x -5. Find domain and range of f(x)and f^-1(x).

f(x) = (3x + 1)/(2x - 5)

Domain :

f(x) is undefined when denominator is zero

==> 2x -5 = 0

==> x = 5/2

Hence Domain is (- , 5/2) U (5/2 , )

Range:

lim _{[x ->}_] f(x) = lim _{[x -> ]} (3x + 1)/(2x - 5)

==> 3/2

Hence range is (- , 3/2) U (3/2 , )

let f(x) = y ==> x = f^-1(y)

==> (3x + 1)/(2x - 5) = y

==> 3x + 1 = y(2x - 5)

==> 3x + 1 = 2xy - 5y

==> 2xy - 3x = 5y + 1

==> x(2y - 3) = 5y + 1

==> x = (5y + 1)/(2y - 3)

==> f ^-1(y) = (5y + 1)/(2y - 3)

==> f ^-1(x) = (5x + 1)/(2x - 3)

Hence Inverse of f(x) = (3x + 1)/(2x - 5) is f ^-1(x) = (5x + 1)/(2x - 3)

Domain :

f ^-1(x) is undefined when denominator is zero

==> 2x -3= 0

==> x = 3/2

Hence Domain is (- , 3/2) U (3/2 , )

Range :

lim _{[x -> -]} f ^-1(x) = lim _{[x -> -]} (5x + 1)/(2x - 3) = 5/2

Hence Range of f ^-1(x) is (- , 5/2) U (5/2 , )