Show that the function f:R→R defined by f(x) = x/(x²+1), ∀x∈R is neither one-one nor onto. Also, if g:R→R is defined as g(x) = 2x-1, find fog(x).

For a function to be one to one, if we assume f(x1) = f(x2), then x1 = x2
Given, f:R→R defined by f(x) = x/(x²+1), ∀x∈R
Thus for one-one function, consider f(x1) = f(x2) ⇒ x1/(x1²+1) = x2/(x2²+1) ⇒ x1(x2²-x1²) = x2-x1 ⇒ x1x2 = 1, if x2 ≠ x1 ⇒ f is not one-one function.
Also, a function is onto if and only if for every y in the co-domain, there is x in the domain such that f(x) = y
Let f(x) = y ⇒ x/(x²+1) = y ⇒ x = y ± √(y²-2y)
Now, substituting this x in f(x) = y we can see that this function is not onto.
Now, fog(x) = f(g(x)) = f(2x-1) = (2x-1)/((2x-1)²+1) = (2x-1)/(4x²-4x+2).