Inverse function

Question

If f:R-R is an invertible function such that f(x) and f^-1(x) are also mirror image to each other about the line y=-x then

 

(A) f(x) is odd.

(B) f(x) and f^-1(x) may not be mirror image about the line y=x

(C) f(x) may not be odd

(D) f(x) is even

Answer 1

 

 Ok  here's an example:

 

     y = f(x) = -2x - 1   is an invertible linear function

 

    y = g(x) = (1-x)/2 is symmetric with f(x) with respect to the line y=-x

 

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By definition, function F(x) is even if F(x) = F(-x); that is, it eats the negative sign

                                F(x) is odd if F(x) = -F(x); that is, it spits out the negative sign

 

 

   f(x) is NOT odd since f(-x) = (-2)(-x) - 1 = 2x - 1   But -f(x) = 2x + 1 so A is not the answer

 

 f(x) and g(x) are not symmetric about the line y=x..... NOT EVEN CLOSE; slopes are negative

  so B is not the answer

 

 f(-x) = 2x-1 as previously shown which is NOT the same as f(x), so f(x) is not even; therefore

  D is not the answer 

 

As shown f(x) is not odd, so the answer is obviously C, f(x) MAY NOT be odd;

Keep in mind however, there MAY be OTHER odd functions that satisfy the symmetry condition y=-x;