Which of the following functions is one-one ? `(1)f:R->R` defined as `f(x)=e^(sgn x)+e^(x^2)` `(2) f:[-1,oo) ->(0,oo)` defined by `f(x)=e^(x^2+|x|)` `(3)f:[3,4]->[4,6]` defined by `f(x)=|x-1|+|x-2|+|x-3|+x-4|` `(4) f(x) =sqrt(ln(cos (sin x))`

Options

A. `f:RrarrR` denined as `f(x)=d^("sgn x")+d^(x^(2))`
B. `f:[-1,oo)rarr(0,oo)` defined by `f(x)=e^(x^(2)+|x|)`
C. `f:[3,4]rarr[4,6]" defined by "f(x)=|x-1|+|x-2|+|x-3|+|x-4|`
D. `f(x)=sqrt(ln(cos(sin x)))`

Answer

C

Explanation

(a) `f(x)=e^(sgnx)+e^(x^(2))`
`{:("when",x=0,"then"f(0)=2),("when",xgt0,"then"f(x)=e+e^(x^(2))),("when",xlt0,"then"f(x)=(1)/(e)+e^(x^(2))):}`

Hence, f(x) is many - one
(b) We have `f(x)=e^(x^(2)+|x|),x in [-1,oo)`
Clearly `f(-1)=e^(2)=f(1)`
also `x^(2)+|x|ge 0 AA x in [-1,oo]`
`rArr" "R_(f)=[1,oo)`
`therefore" "f(x)` is many-one into function.
`f(x)=|x-1|+|x-3|+|x-4|,x in [3, 4]` ltBrgt `=(3x-6)-(x-4)`
`f(x)=2x-2`, which is increasing function
`R_(f)=[4,6]`
Clearly, f(x) is one-one onto function.
`f(x)=sqrt(ln(cos(sinx)))`
For domain, `ln(cos(sinx))ge0`
`rArr" "(cos(sinx))ge1`
`rArr" "cos(sinx)=1,`
`therefore" "sinx=0`
`rArr" "x=npi, n in I`
`" "R_(f)={0}`
since `f(x)=0`
Thus, f(x) is many-one function.