Inverse of function WITH a limit??

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October 30th 2008, 12:37 PM
mwok

Inverse of function WITH a limit??

Find f^-1(x) <-- inverse if f(x) = x^2 - 4x + 3, x <= 2 find value of f^-1(8).

I got x=29 BUT x<= 2????
October 30th 2008, 12:53 PM
masters

Quote:

Originally Posted by mwok

Find f^-1(x) <-- inverse if f(x) = x^2 - 4x + 3, x <= 2 find value of f^-1(8).

I got x=29 BUT x<= 2????

What did you find your inverse function to be?

Is it $f^{-1}(x)=\pm\sqrt{x+1}+2$ ?
October 30th 2008, 01:00 PM
mwok

Quote:

Originally Posted by masters

What did you find your inverse function to be?

Is it $f^{-1}(x)=\pm\sqrt{x+1}+2$ ?

I just replaced x with y so my inverse function became

inverse(x) = y^2 - 4y + 3

which yield 29....is that correct? can't be cause x should be less than 2 (or equal to)
October 30th 2008, 01:12 PM
masters

Quote:

Originally Posted by mwok

I just replaced x with y so my inverse function became

inverse(x) = y^2 - 4y + 3

which yield 29....is that correct? can't be cause x should be less than 2 (or equal to)

Not quite. Your original function was $f(x)=x^2-4x+3$

$f(x)=y=x^2-4x+3$

Interchange x and y

$x=y^2-4y+3$

Isolate the y

$y^2-4y=x-3$

Complete the square on the left side. You'll see why in a minute.

$y^2-4y+4=x-3+4$

$(y-2)^2=x+1$

$y-2=\pm\sqrt{x+1}$

$y=\pm\sqrt{x+1}+2$

$\boxed{f^{-1}(x)=\pm\sqrt{x+1}+2}$
October 30th 2008, 01:18 PM
mwok

Wow! I didn't know you could plus 4 to each side and solve the problem like that. Are there multiple ways to solve this?
October 30th 2008, 01:28 PM
masters

Quote:

Originally Posted by mwok

Wow! I didn't know you could plus 4 to each side and solve the problem like that. Are there multiple ways to solve this?

You could've used the quadratic formula, but completing the square seems to be the best way.

Now, can you find $f^{-1}(8)$ ?
October 30th 2008, 01:30 PM
mwok

Quote:

Originally Posted by masters

You could've used the quadratic formula, but completing the square seems to be the best way.

Now, can you find $f^{-1}(8)$ ?

Yep, got it. It can either be -1 or 5 BUT since x should be less or equal to 2, it is automatically -1.

Thanks so much!