anonymous
  • anonymous
Let f(x)=10x-10. Find the value of (f o f^-1)(-10)
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
f(x) is a one-to-one function (it's a line) so (f o f^-1)(-10) = -10
anonymous
  • anonymous
basically, inverses "undo" each other, so you end up with the same value
anonymous
  • anonymous
oh ok and so i plug the -10 into the original equation or would that be the answer?

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Jhannybean
  • Jhannybean
Nope.\[f(f^{-1}(-10)) = x\]\[-10=x\] That's basically what is going on. \[f(-10)=-10x-10 = -110 \ne -10 \]
Jhannybean
  • Jhannybean
Sorry, forgot to substitute by -10 in for x. \(f(-10)=-10(-10)-10 = -110\)*
UnkleRhaukus
  • UnkleRhaukus
\[f(x) = 10x-10\\[3ex] x=10\Big(f^{-1}(x)\Big)-10\\ x+10 = 10(f^{-1}(x))\\ f^{-1}(x)=\frac {x+10}{10}\\[3ex]f\circ f^{-1}(x)=f\Big(\frac {x+10}{10}\Big)\\ \qquad\qquad\quad = 10\Big(\frac {x+10}{10}\Big)-10\\ \qquad\qquad\quad = x+10-10\\ \qquad\qquad\quad =x\\[2ex] f\circ f^{-1}(-10) =-10\]
anonymous
  • anonymous
\[f \circ f^{-1}(x)=x\] A function and it's inverse cancel each other out, leaving you with the independent variable x
Jhannybean
  • Jhannybean
I like the way @UnkleRhaukus proved it and then found the value.
anonymous
  • anonymous
mmhm, it has a beauty of it's own
anonymous
  • anonymous
yes it does

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