Discussion Forum

1)

Let  $f:R\rightarrow R $ and $g:R\rightarrow R$  be the functions defined by  $f(x)= \frac{x}{1+x^{2}}$,

$x \in R, g(x)=\frac{x^{2}}{1+x^{2}},x\in R$  Then, the correct statement (s) among the following is/are

(a) both f.g are one-one

(b)  both f.g are onto

(c)  both f.g are not one-one  as well as onto

(d) f and g are onto but not one-one

Answer:

Option D

Explanation:

 We have,

 $f(x)= \frac{x}{1+x^{2}}$  $x \in R$

$g(x)=\frac{x^{2}}{1+x^{2}},x\in R$

$f'(x)= \frac{1+x^{2}-2x^{2}}{(1+x^{2})^{2}}$

$f'(x)= \frac{1-x^{2}}{(1+x^{2})^{2}}$

 Clearly  f'(x)  is not monotonic

$\therefore$    f(x)  is not one-one function range of f(x) , is    $\left[ -\frac{1}{2},\frac{1}{2}\right]$

 $\therefore$   f(x) is not onto

 Clearly g(x) is even function

 $\therefore$   g(x)  is not one-one function 

Range of g(x) is [0,1]

 $\therefore$   g(x) is also not onto . Here , f(x) and g(x) both are neither one-one not onto.