Question

Let f(x)=3 ×^2 tan((pi)(x)/2), where -1 <×<1 a) find f^-1 (3) b) find f (f^-1 (5))

Answer 1

`f(x) = 3x^(2)tan((\pi x)/(2))`

`f(x) = 3x^(2)[(1 - cos(\pi x))/(sin(\pi x))]`

`f(x) = 3x^(2)[csc(\pi x) - cot(\pi x)]`

`f(x) = 3x^(2)csc(\pi x) - 3x^(2)cot(\pi x)`

`y = 3x^(2)csc(\pi x) - 3x^(2)cot(\pi x)`

`x = 3y^(2)csc(\pi x) - 3y^(2)cot(\pi x)`


`x = 3y^(2)csc(\pi x) - 3y^(2)cot(\pi x)`

`3 = 3y^(2)csc(\pi x) - 3y^(2)cot(\pi x)`

`3 = 3y^(2)[csc(\pi x)] - 3y^(2)[cot(\pi x)]`

`3 = 3y^(2)[csc(\pi x) - cot(\pi x)]`

`3 = 3y^(2)[(1 - cos(\pi x))/(sin(\pi x))]`

`3 = 3y^(2)tan((\pi)/(2)y)`

`1 = y^(2)tan((\pi)/(2)y)`

`(1)/(tan((\pi)/(2)y)) = y^(2)`

`cot((\pi)/(2)y) = y^(2)`

`cot^(-1)[cot((\pi)/(2)y)] = cot^(-1)(y^(2))`

`(\pi)/(2)y = cot^(-1)(y^(2))`

`\pi y = 2cot^(-1)(y^(2))`

`y = (2cot^(-1)(y^(2)))/(\pi)`


`x = 3y^(2)csc(\pi x) - 3y^(2)cot(\pi x)`

`5 = 3y^(2)csc(\pi x) - 3y^(2)cot(\pi x)`

`5 = 3y^(2)[csc(\pi x)] - 3y^(2)[cot(\pi x)]`

`5 = 3y^(2)[csc(\pi x) - cot(\pi x)]`

`5 = 3y^(2)[(1 - cos(\pi x))/(sin(\pi x))]`

`5 = 3y^(2)tan((\pi)/(2)y)`

`1(2)/(3) = y^(2)tan((\pi)/(2)y)`

`(5)/(3tan((\pi)/(2)y)) = y^(2)`

`1(2)/(3)cot((\pi)/(2)y) = y^(2)`

`\sqrt{1(2)/(3)cot((\pi)/(2)y)} = y`