Question

If f(x)= $sin^{-1} ( x^{2}/1+ x^{2} )$
then find range of f(x) is?

Answer 1

let f(x) = y 

y = sin⁻¹ {x²/1+x²}

since x²/1+x² can never be negative 

so siny = x²/1+x² 

⇒ x = $\sqrt{ \frac{siny}{1-siny} }$

so 1-siny > 0 ⇒ 1 > siny 

⇒ y < π/2

again $\frac{siny}{1-siny}$ ≥ 0

⇒ ${siny}{1-siny}$ ≥ 0           (since 1- siny can never be negative )

⇒  siny ≥ 0 ⇒ y ≥ 0 
 
so plotting in graph and evaluating by wave method we get y ∈ [0,π/2) 

which is the required range