Question

Prove that 𝑓(𝑥) = 𝑥 2 + 2𝑥 is not injective

Answer 1

Given,

$f(x)=x^2+2x$

$x^2+2x-y=0$

The solution of the above equation is-

$x=\dfrac{-2\pm\sqrt{4+4y}}{2}$

$x=-1\pm\sqrt{1+y}$

which is defined in R for $y≥−1.$ This can be used to separate the domain of f(x) into two intervals. For $x\in[1,∞$ ) and $x\in(−∞,1]$ , the function f(x) is not injective (and invertible). The domain that contains 2 is of course $[1,∞).$