f: X arrow C, X=\{x 0

Correct Answer is : fff is bijective

f is neither injective nor surjective

Correct Answer is : fff is bijective

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WBJEE 2022 Math Solved Paper

Question : 64 of 75

Marks: +1, -0

f: X \rightarrow C, X=\{x \mid 0 < x < 1\}

f(x)=\frac{2x-1}{1-|2x-1|}

. Then

Solution:

Hint : Put

2x-1=t

, then the function becomes

f(t)=\frac{t}{1-|t|}, -1 < t < 1

\therefore f(t)=\begin{cases} \frac{t}{1+t}, & -1 < t \leq 0 \\ \frac{t}{1-t}, & 0 < t < 1 \end{cases}

\because

It is continuous and

f(-1^{+})=-\infty, f(1^{-})=+\infty

\therefore \text{Range}=(-\infty,-\infty)=C

It is differentiable also,

f'(t)=\begin{cases} \frac{1}{(1+t)^{2}}, & -1 < t < 0 \\ \frac{1}{(1-t)^{2}}, & 0 < t < 1 \end{cases}

\because f'(t) > 0 \quad \forall -1 < t < 1

\therefore f

is injective

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