Determine whether the function is one-to-one. The domain is the set of all real

Question

Determine whether the function is one-to-one. The domain is the set of all real numbers. If the function is not one-to-one, exhibit distinct numbers a and b with f(a)=f(b). Also, determine if the function is onto, exhibit a number y for which f(x) does not equal y for all real x.

f(x) = x / (1+x^2)

Explanation / Answer

let x1 = x2 then x1^2 = x2^2 => 1+ x1^2 = 1+ x2^2 => x1 / (1+x1^2) = x2 / (1+ x2^2) => f(x1) = f(x2) hence the function is one - one. Take y = 1 Inverting f(x) we get x+ 1/x and x+(1/x) is always greater than 2 hence f(x) is always less than 1/2

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