Question

The graph of the function $f\left(x\right)=\frac{x}{1+x^2}$ is given as:

Then which of the following is/are true regarding $f\left(x\right)?$

• A. $f\left(x\right)$ is one-one function
• B. $f\left(x\right)$ is many-one function
• C. Range of $f\left(x\right)$ is $[\frac{-1}{2},\frac{1}{2}]$
• D. $f\left(x\right)$ is one-one function in $x\in [1,\infty )$

Answer 1

Answer: (B), (C), (D)

$f\left(x\right)$ is one-one function in $x\in [1,\infty )$

Clearly using horizontal line test, we can say that the given function is many-one.

In the interval $x\in [1,\infty )$, clearly it is a one-one.

The graph of $f\left(x\right)$ takes all the values in $[\frac{-1}{2},\frac{1}{2}]$
Hence, the range is $[\frac{-1}{2},\frac{1}{2}].$