Question

Let
$f\left(x\right)=\{\begin{array}{cc}x\sin \left(\frac{1}{x}\right)& \text{when}x\ne 0\\ 1& \text{when}x=0\end{array}$
and $A=\{x\in R:f(x)=1\}$. Then $A$ has

• A. exactly one element
• B. exactly two elements
• C. exactly three elements
• D. infinitely many elements

Answer 1

The correct option is A exactly one element

$f\left(x\right)=1$ when $x=0$
When $x\ne 0$
$f\left(x\right)=1\Rightarrow x\sin \left(\frac{1}{x}\right)=1\Rightarrow \sin \left(\frac{1}{x}\right)=\frac{1}{x}$
assuming $\frac{1}{x}=z$
$\sin z=z$

the only possible solution is $z=0$
$\Rightarrow \frac{1}{x}=0$ which is not possible
$\therefore x=0$ is only solution
Hence, only one element in $A$.