function inverse

red and white kop!

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Jun 15, 2009
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f(x) = x^2 - 8x
find f^-1 (x)
i know this must be very simple but i have no idea what to do here. can somebody just show me how it is done? i am really at a loss of what to do, especially since i usually have no problem with inverse functions. i just use f(x)= y and solve for x, then replace y with x.
 
red and white kop! said:
f(x) = x^2 - 8x
find f^-1 (x)
i know this must be very simple but i have no idea what to do here. can somebody just show me how it is done? i am really at a loss of what to do, especially since i usually have no problem with inverse functions. i just use f(x)= y and solve for x, then replace y with x.

That's right. Folowing your steps, you get : y = x[sup:31ysa5fh]2[/sup:31ysa5fh]-8x, that is, x[sup:31ysa5fh]2[/sup:31ysa5fh]-8x-y=0.

You need to use the quadratic formula to solve this :

ax2+bx+c=0\displaystyle ax^2+bx+c=0 gives x=b±b24ac2a\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

Here, that means

x=8±64+4y2\displaystyle x=\frac{8\pm\sqrt{64+4y}}{2}

Exchanging x and y gives :

y=8±64+4x2\displaystyle y=\frac{8\pm\sqrt{64+4x}}{2}

Going back to x=8±64+4y2\displaystyle x=\frac{8\pm\sqrt{64+4y}}{2}, this means deciding whether your original function f(x) is defined on a sub-interval of x4\displaystyle x\geq4 or of x4\displaystyle x\leq 4. If it's defined everywhere, it doesn't have an inverse function.
 
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