Show that the function $f:R \rightarrow R$ defined by $f(x)=\dfrac{x}{x^{2}+1}$, $\forall x \in R$, is neither one-one nor onto. Also, if $g:R \rightarrow R$ is defined as $g(x)=2x-1$, find $fog(x)$.

$f$ is neither one-one nor onto; $fog(x)=\dfrac{2x-1}{4x^{2}-4x+2}$

CBSE Class 12 · 2018 · Relations And Functions XIILong Answer

Show that the function

f : R \to R

defined by

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

\forall x \in R

, is neither one-one nor onto. Also, if

g : R \to R

is defined as

g (x) = 2 x - 1

, find

f o g (x)

Answer: $f$ is neither one-one nor onto; $f o g (x) = \frac{2 x - 1}{4 x ^{2} - 4 x + 2}$

Not one-one: take

x = 2

and

x = \frac{1}{2}

f (2) = \frac{2}{5}, f (\frac{1}{2}) = \frac{1/2}{1/4 + 1} = \frac{2}{5}

Equal images from distinct inputs (

2 \neq = \frac{1}{2}

), so

f

is not one-one.
Not onto: try to attain

y = \frac{1}{2}

. Solving

\frac{x}{x ^{2} + 1} = \frac{1}{2}

gives

x^{2} - 2 x + 1 = 0

, whose discriminant is

D = (- 2)^{2} - 4 (1) (1) = 2 - 4 = - 2 < 0

so there is no real pre-image;

f

is not onto.
Finally,

f o g (x) = f (2 x - 1)

f o g (x) = \frac{2 x - 1}{( 2 x - 1 ) ^{2} + 1} = \frac{2 x - 1}{4 x ^{2} - 4 x + 2}

f is neither one-one nor onto; f o g (x) = \frac{2 x - 1}{4 x ^{2} - 4 x + 2}

CBSE Class 12 Board · 2018 · Subjective