Question

Which of the following functions is/are injective map(s)?

• A. $f\left(x\right)=x^2+2,x\in (-\infty ,\infty )$
• B. $f\left(x\right)=|x+2|,x\in [-2,\infty )$
• C. $f\left(x\right)=(x-4)(x-5),x\in (-\infty ,\infty )$
• D. $f\left(x\right)=\frac{4x^2+3x-5}{4+3x-5x^2},x\in (-\infty ,\infty )$

Answer 1

The correct option is B $f\left(x\right)=|x+2|,x\in [-2,\infty )$

(A) The function $f\left(x\right)=x^2+2,x\in (-\infty ,\infty )$ is not injective as

$f\left(1\right)=f(-1)$ but $1\ne -1$

(C) The function $f\left(x\right)=(x-4)(x-5),x\in (-\infty ,\infty )$ is not one-one as

$f\left(4\right)=f\left(5\right)$ but $4\ne 5$

(D) The function $f\left(x\right)=\frac{4x^2+3x-5}{4+3x-5x^2},x\in (-\infty ,\infty )$ is also not injective as

$f\left(1\right)=f(-1)$ but $1\ne -1$

(B) For the function $f\left(x\right)=|x+2|,x\in [-2,\infty )$
Let $f\left(x\right)=f\left(y\right),x,y\in [-2,\infty )\Rightarrow |x+2|=|y+2|$
$\Rightarrow x+2=y+2\Rightarrow x=y$
So, This is an injection.