INVERSE FUNCTIONS

Question

Find f −1(a)  for the function f and the real number a.    '

f(x)= (9/π)x+sinx ;   −π/2 < x < π/2   ; a = 2

f −1(a) =????

 

 

Thanks

Answer 1

f(x)= (9/pi)x+sinx

 

One would be hard pressed to come up with an explicit formula for the inverse of f.

 

If (z,a) is a point on f, so that f(z) = a, then f^-1(a) = z, based on the idea of interchanging domain & range to get inverse, but I don't see how to solve  f(z) = 2 for z.

 

Answer 2

f(x) = (9/π)x + sinx

 

If f^-1(2) = b, then f(b) = 2.

 

So, we are looking for the value of b in [-π/2, π/2] such that f(b) = 2.

 

By inspection, f(π/6) = (9/π)(π/6) + sin(π/6)

 

                              = 3/2 + 1/2 = 2

 

Therefore, f^-1(2) = π/6