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Question
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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Solution
For x1 , x2 ∈ R, consider
f(x1) = f(x2)
⇒ `x_1/(x_1^2 + 1) = x_2/(x_2^2 + 1)`
⇒ `x_1 x_2^2 + x_1 = x_2 x_1^2 + x_2`
⇒ x1 x2 (x2 – x1) = x2 – x1
⇒ x1 = x2 or x1 x2 = 1
We note that there are point, x1 and x2 with x1 ≠ x2 and if f(x1) = f(x2), for instance, If we take x1 = 2 and x2 = `1/2`, then we have f(x1) = `2/5` and f(x2) = `2/5` but `2 ≠ 1/2`.
Hence f is not one-one. Also, f is not onto for if so then for 1∈R ∃ x ∈ R such that f(x) = 1 which gives `x/(x^2 + 1)` = 1
But there is no such x in the domain R, since the equation x2 – x + 1 = 0 does not give any real value of x.
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