Question

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

Answer 1

We have , $f(x)=x^2+2x$

$Calculate f(x_1):\\ \Rightarrow f(x_1)=x_1^2+2x\\ Calculate f(x_2):\\ \Rightarrow f(x_2)=x_2^2+2x$

Suppose the function is injective then $f(x_1)=f(x_2)$

if this condition is not true then function is not injective

Now, $f(x_1)=f(x_2)$

$\Rightarrow x_1^2+2x=x_2^2​+2x​$

$\Rightarrow x_1^2-x_2^2​+2x​_1-2x_2=0\\ \Rightarrow (x_1−x_2)(x_1​+x_2)+2x_1−2x_2=0\\ \Rightarrow (x_1​−x_2)(x_1​+x_2+2)=0\\$

Since, $x_1\neq x_2$ and $x_1​+x_2​+2\neq0 ,$ for any $x∈N$

So function f(x) is not injective.