The range of the real valued function f(x)= ⁻¹ ( 1+x^2 2 x )+ ⁻¹ ( 2 x 1+x^2 ) is

Step-by-step solution

f(x)= ⁻¹ ( 1+x^2 2 x )+ ⁻¹ ( 2 x 1+x^2 ) We know, domain of ⁻¹(x) is [-1,1] -1 1+x^2 2 x 1 and -1 2 x 1+x^2 1 When, -1 1+x^2 2 x 1 1+x^2 2 x 1 and 1+x^2 2 x -1 aligned & (1-x)^2 2 x 0 and (1+x)^2 2 x 0 & x 0, x=1 and x 0, x=-1 x=-1,1...(i) aligned aligned & When, -1 2 x 1+x^2 1 & 2 x 1+x^2 1 and 2 x 1+x^2 -1 & -(1-x)^2 1+x^2 0 and (1+x)^2 1+x^2 0 x R ...(ii) aligned From equations (i) & (ii), x=-1,1 ; Domain = -1,1 and f(1)= ⁻¹ ( 1+1 2 )+ ⁻¹ ( 2 1+1 )= 2 f(-1)= ( 1+1 -2 )+ ⁻¹ ( -2 1+1 )=- 2 + = 2 Range (f)= 2