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

Answer

N/a

Explanation

Domin of f (x) :
f (x) is defined for all real values of x.
`therefore` Domin of f(x) = R.
Range of f (x):
Let y = f (x)
`implies" "y=(x)/(1+x^(2))`
`" "y+x^(2)y=x`
`implies" "x^(2)y-x+y=0`
`implies" "x=(a+-sqrt(1-4y^(2)))/(2y)`
`therefore` x is defined if
`1-4y^(2)ge0andyne0`
`implies(1-2y)(1+2y)ge0andyne0`
`implies" "y in[-(1)/(2),(1)/(2)]-{0}`
But at x = 0, y=0
`therefore"Range of"f(x)=[-(1)/(2),(1)/(2)].`