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How to find the inverse function?

We are given the function y = square root of (x2 - 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 (x2 - 6x + 15)
= square root of (x2 - 6x + 32 - 32 + 15)
= square root of (x2 - 6x + 32 - 9 + 15)
= square root of (x2 - 6x + 32 + 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]
y2 = (x - 3)2 + 6
now express x in terms of y
(x - 3)2 = y2 - 6
x - 3 = either positive or negative of square root of (y2 - 6)
since x is <= 3 (given), so (x - 3) <= 0
so x = 3 = - square root of (y2 - 6) and negative infinity < y <= square root of 6.
now change y to x and change x to y
y = 3 - square root of (x2 - 6) and negative infinity < x <= square root of 6.
so we get the inverse function
f-1(x) = 3 - square root of (x2 - 6) and negative infinity < x <= square root of 6.