Find the domain and range of the function `f(x) = x/(1+x^2)`

Explanation

We have, `f(x)=(x)/((1+x^(2)))`.
Clearly, f(x) is defined for all `x""inR`. So, dom (f)=R.
Let y=f(x). Then,
`y=(x)/((1+x^(2)))impliesx^(2)y-x+y=0`
`impliesx=(1+-sqrt(1-4y^(2)))/(2y)." "......(i)`
It is clear from (i) that x will take real values, when
`(1-4y^(2))ge0andyne0`
`implies(4y^(2)-1)le0andyne0`
`implies(2y+1)(2y-1)le0andyne0`
`implies(y+(1)/(2))(y-(1)/(2))le0andyne0`
`implies-(1)/(2)leyle(1)/(2)andyne0`
`impliesyin[(-1)/(2),(1)/(2)]-{0}`
Also, `x=0impliesy=0`.
`:."range "(f)=[(-1)/(2),(1)/(2)]`.
Hence, dom (f)=R and range `(f)=[(-1)/(2),(1)/(2)]`.