Inverse trig question - Cambridge (1 Viewer)

[pwoh]

Yr 12 Cambridge, 1A, Question 20.






The answer is but I don't understand why.

Also, how would you: "Prove, both geometrically and algebraically, that if an odd function has an inverse function, then that inverse function is also odd."?

Thanks

 

[Trebla]

Also, how would you: "Prove, both geometrically and algebraically, that if an odd function has an inverse function, then that inverse function is also odd."?

Thanks

For the algebraic proof, if a function y = f(x) is odd which has an inverse then
f(- x) = - f(x)
=> - y = f(- x)
=> - x = f^-1(- y)
But x = f^-1(y) since y = f(x) hence
- f^-1(y) = f^-1(- y)
Through the interchange of x and y this gives
- f^-1(x) = f^-1(- x)
which implies y = f^-1(x) is an odd function

 

[Drongoski]

y = f(x) = (x² -4x + 24)/6 being non-monotone has no inverse; if we consider negative x, the left branch of this parabola is monotone and has an inverse.

 

[pwoh]

Thanks guys, much appreciated

If I were to prove " that if an odd function has an inverse function, then that inverse function is also odd" geometrically, how should I word it?

I'm thinking something about rotational symmetry about the origin and flipping across y=x, but I don't know how to put it together.