Most Important Selected Qs for JEE AdvancedMathematicsFunctions
Which of the following function(s) is(are) surjective?
Options
- Af: D_f R, f(x)= ( ( [x])+ |x^2+2 x-3 | )
- Bg: D_g R, g(x)= x^2+2 x-3 x-1
- Ch: D_h R, h(x)= ( 1-x 1+x )
- Dk: D_k R⁺, k(x)= [x]+[-x]+1 + x + -x +1 [Note: [m] and m denotes greatest integer function less than or equal
Correct answer
C. h: D_h R, h(x)= ( 1-x 1+x )
Step-by-step solution
() aligned & f(x)= ( [x]+ |x^2+2 x-3 | ) & [x] I [x]=0, & and |x^2+2 x-3 |= |(x+1)^2-2 | [0, ) aligned So, f(x) R f(x) is surjective. (2) aligned & g(x)= x^2+2 x-3 x-1 , x 1 & g(x)= (x-1)(x+3) (x-1) , x 1 & g(x)=x+3 g(x) 4 ( x 1) aligned So, range of g(x) is R- 4 . g(x) is not surjective. (3) h(x)= ( 1-x 1+x ), 1-x 1+x >0 1-x 1+x take all value between (0, ) So, range of h(x)=R h(x) is surjective. (4) k(x) [x]+[-x]+1 + x + -x +1 Domain of k(x) is R aligned & x I and x +[-x]=-1 -x =1 & k(x)= 2 & x I [x]+[-x]=0 and x