Functions 3 Question 7
7. Which of the following functions is periodic? $(1983,1 M)$
(a) $f(x)=x-[x]$, where $[x]$ denotes the greatest integer less than or equal to the real number $x$
(b) $f(x)=\sin (1 / x)$ for $x \neq 0, f(0)=0$
(c) $f(x)=x \cos x$
(d) None of the above
Objective Question II
(One or more than one correct option)
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Answer:
Correct Answer: 7. (c)
Solution:
- We have, $f(x)=\frac{x}{1+x^{2}}$
$$ \therefore \quad f \frac{1}{x}=\frac{\frac{1}{x}}{1+\frac{1}{x^{2}}}=\frac{x}{1+x^{2}}=f(x) $$
$\therefore \quad f \frac{1}{2}=f(2)$ or $f \frac{1}{3}=f(3)$ and so on.
So, $f(x)$ is many-one function.
Again, let $\quad y=f(x) \Rightarrow y=\frac{x}{1+x^{2}}$
$\Rightarrow \quad y+x^{2} y=x \Rightarrow y x^{2}-x+y=0$
As, $\quad x \in R$
$\therefore \quad(-1)^{2}-4(y)(y) \geq 0$
$\Rightarrow \quad 1-4 y^{2} \geq 0$
$\Rightarrow \quad y \in \frac{-1}{2}, \frac{1}{2}$
$\therefore$ Range $=$ Codomain $=\frac{-1}{2}, \frac{1}{2}$
So, $f(x)$ is surjective.
Hence, $f(x)$ is surjective but not injective.
