Find the domain and range of the function `f(x)=(x^2)/(1+x^2)` . Is the function one-to-one?

Options

A. `[0,1//2]`
B. `[-1//2,1//2]`
C. `[-1//2,0]`
D. `[-1//2,0) cup (0,1//2]`

Answer

B

Explanation

Clearly, f(x) is defined for all ` x in R. `
So, domain (f)=R
In order to find the range of f(x) , let
y=f(x)
`implies y=(x)/(x+x^(2)`
`implies x^(2)y-x+y=0`
`implies x=(1 pm sqrt(1-4y^(2)))/(2y)`
For x to be real, we must have
`1-4y^(2) ge 0 and y ne 0 implies -(1)/(2) le y le (1)/(2) and y ne 0`
Also , y=-0 for x=0
Hence, range of f(x)=[-1/2,1/2]