Solve functional equation f ( x ) + f ( y ) = f ( x 1 − y 2 + y 1 − x 2 )

Question

Solve functional equation   f  (  x  )  +  f  (  y  )  =  f  (  x      1    −          y      2        +  y      1    −          x      2        )

motylowceyvy · Accepted Answer

x  =  y  =  0    ⟹    f  (  0  )  =  0  x  =  y  =  1    ⟹    2  f  (  1  )  =  f  (  0  )  =  0    ⟹    f  (  1  )  =  0  x  =  sin  ⁡  t  ,    y  =  cos  ⁡  t    ⟹    f  (  sin  ⁡  t  )  +  f  (  cos  ⁡  t  )  =  f  (  1  )  =  0  x  =  y  =  cos  ⁡  t    ⟹    2  f  (  cos  ⁡  t  )  =  f  (  2  sin  ⁡  t  cos  ⁡  t  )  x  =  y  =  sin  ⁡  t    ⟹    2  f  (  sin  ⁡  t  )  =  f  (  2  sin  ⁡  t  cos  ⁡  t  )Hence,   f  (  sin  ⁡  t  )  =  f  (  cos  ⁡  t  )  =  −  f  (  sin  ⁡  t  ) and so the only solution is f=0.

Solve functional equation f(x)+f(y)=f(x sqrt(1−y^2)+y sqrt(1−x^2))

Answered question

Answer & Explanation

New Questions in Trigonometry