We are given the function y = square root of (x^{2} - 6x + 15), the domain of the function is x <= 3 and x > negative infinity,
The question ask for the inverse function f^{-1}(x). Now we express the function in maximum form.

- y = square root of (x
^{2}- 6x + 15) - = square root of (x
^{2}- 6x + 3^{2}- 3^{2}+ 15) - = square root of (x
^{2}- 6x + 3^{2}- 9 + 15) - = square root of (x
^{2}- 6x + 3^{2}+ 6) - = square root of [(x - 3)
^{2}+ 6]

Now we will find the range of y from the domain of the function. When x tend to negative infinity, y tend to positive infinity. When x = 3, y equals to the square root of 6. So the range of y is y >= square root of 6 and y < positive infinity.

Now we are going to find the inverse function f^{-1}(x).

- y = square root of [(x - 3)
^{2}+ 6] - y
^{2}= (x - 3)^{2}+ 6 - now express x in terms of y
- (x - 3)
^{2}= y^{2}- 6 - x - 3 = either positive or negative of square root of (y
^{2}- 6) - since x is <= 3 (given), so (x - 3) <= 0
- so x = 3 = - square root of (y
^{2}- 6) and negative infinity < y <= square root of 6. - now change y to x and change x to y
- y = 3 - square root of (x
^{2}- 6) and negative infinity < x <= square root of 6. - so we get the inverse function
- f
^{-1}(x) = 3 - square root of (x^{2}- 6) and negative infinity < x <= square root of 6.