How do I find the inverse f(x)= 24x+5/ (x-120)   and x ≠ 4? ?

f(x) = (24x + 5)/(x - 120)

y = (24x + 5)/(x - 120) → to obtain the inverse, you substitute x by y and you substitute y by x

x = (24y + 5)/(y - 120)

x.(y - 120) = 24y + 5

xy - 120x = 24y + 5

xy - 24y = 120x + 5

y.(x - 24) = 120x + 5

y = (120x + 5)/(x - 24)

y = 5.(24x + 1)/(x - 24) ← this is the inverse of the function f

Let x = f(y) = 24y + 5/(y - 120)

=> xy - 120x = 24y^2 - 2880y + 5 =>

use quadratic formula to solve for y and note that y = f^(-1)(x), but use only the positive square root, as f has only a partial inverse.

On the other hand, if you meant f(x) = (24x + 5)/(x - 120), that's a different story.

Let x = f(y) =>

xy - 120x = 24y + 5 =>

(x - 24)y = 120x + 5 =>

y = (120x + 5)/(x - 24) = the inverse of f(x).

...........24x + 5

f(x) = -----------------

............. x - 120

..............24x + 5

y = ----------------------

..........x - 120

........... 24y + 5

x = -------------------

...............y - 120

x(y - 120) = 24y + 5

 xy - 120x = 24y + 5

 xy - 24y = 120x + 5

y(x - 24) = 120x + 5

...........120x + 5.............. 5(24x + 1)

y =  -------------------- or --------------------- Answer//

..............x - 24....................x - 24