Find the inverse of f if f(x) = x^2 - 8x + 8 and x is less than or equal to 4

[ironspud]

Homework Statement

Find $f^{-1}(x)$ if $f(x)=x^{2}-8x+8$ and $x\leq4$

The Attempt at a Solution

I set $y=x^{2}-8x+8$, and then switch y and x to get $x=y^{2}-8y+8$.
I then try solving for y, but I end up with y's on both sides of the equation:

$x=y^{2}-8y+8$

$x-8=y^{2}-8y$

$x-8=y(y-8)$

$\frac{x-8}{y-8}=y$

$?$

 

[CompuChip]

It's a quadratic equation in y, so try the quadratic formula :-)

 

[SammyS]

or complete the square.

$x-8=y^{2}-8y$

$x-8+16=y^{2}-8y+16$

$x+8=(y-4)^2$

   ⋮

Don't forget the ± when taking the square root.

The range of a function's inverse, f ^-1(x), is the same as the domain of the function, f(x).