Can the inverse of a function be the same

When finding the inverse for f(x)=(x+5)/(2x+3), I got f^-1(x)=(x+5)/(2x+3).

I noticed that they are the same. Is this normal or did I do something wrong?

They are the same in as much as an image in a mirror is the same as it's reflection. When graphed, the inverse of a function is reflected across the line y = x, so the only function whose inverse is the same as itself would be, well, y = x.

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

Inverting f(x),

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

Subbing fˉ¹(x) for y,

fˉ¹(x) = (5 - 3x) / (2x - 1)

If you plot f(x), fˉ¹(x) and y = x, you will see the mirror image. 
 

Others all ready provided a detailed answer, but what I wanted to add is that math is not something you guess whether or not you are correct. As you learn new concepts, there are always ways to check whether or not you are correct based on their definitions. For instance, an inverse function f^{-1}(x), is defined by the following property:

f(f^{-1}(x)) = x

If you put in the inverse you got wherever x is, the result does not simplify to just "x" as it should given the definition of an inverse. Try to put in the inverse given by the other answerers, you will see that you get x.

You did something wrong.
.. y = (x+5)/(2x+3)
.. y(2x+3) = (x+5)
.. x(2y-1) = (5-3y)
.. x = (5-3y)/(2y-1)
.. f^-1(x) = (5-3x)/(2x-1)