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 asks 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 tends to negative infinity, y tends 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.