Question

Please find the inverse function of f.

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

Answer 1

What is the domain of f? What is the domain of the inverse function?

Answer 2

If 2x-5 = 0, we have a problem
then x = 5/2
so domain is all x except x = 5/2

(2x-5)y = (x+3)
2 x y - 5 y = x + 3
x(2y-1) = 5y+3
f(y) = (5y+3)/(2y-1)
This one blows up when y = 1/2

Answer 3

Let y = f(x)

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

Switch x and y.

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

Multiply both sides by (2y - 5).

2xy - 5x = y + 3

Subtract 3 from both sides.

2xy - 5x - 3 = y

Subtract 2xy from both sides.

-5x - 3 = -2xy + y

Factor the right side.

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

Divide both sides by (-2x + 1).

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

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

Now replace y with the inverse function notation: f^(-1) x.

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