Question
Which of the following functions is (are) injective map(s) ?
A.
$$f\left( x \right) = {x^2} + 2,\,x\, \in \left( { - \infty ,\,\infty } \right)$$
B.
$$f\left( x \right) = \left| {x + 2} \right|,\,x\, \in \left[ { - 2,\,\infty } \right)$$
C.
$$f\left( x \right) = \left( {x - 4} \right)\left( {x - 5} \right),\,x\, \in \left( { - \infty ,\,\infty } \right)$$
D.
$$f\left( x \right) = \frac{{4{x^2} + 3x - 5}}{{4 + 3x - 5{x^2}}},\,x\, \in \left( { - \infty ,\,\infty } \right)$$
Answer :
$$f\left( x \right) = \left| {x + 2} \right|,\,x\, \in \left[ { - 2,\,\infty } \right)$$
Solution :
The function $$f\left( x \right) = {x^2} + 2,\,x\, \in \,\left( { - \infty ,\,\infty } \right)$$ is not injective as $$f\left( 1 \right) = f\left( { - 1} \right){\text{ but }}1 \ne - 1.$$
The function $$f\left( x \right) = \left( {x - 4} \right)\left( {x - 5} \right),\,x\, \in \left( { - \infty ,\,\infty } \right)$$ is not one-one as $$f\left( 4 \right) = f\left( 5 \right),{\text{ but}}\,4 \ne 5.$$
The function, $$f\left( x \right) = \frac{{4{x^2} + 3x - 5}}{{4 + 3x - 5{x^2}}},\,x\, \in \left( { - \infty ,\,\infty } \right)$$ is also not injective as $$f\left( 1 \right) = f\left( { - 1} \right),{\text{ but }}1 \ne - 1$$
For the function, $$f\left( x \right) = \left| {x + 2} \right|,\,x\, \in \left[ { - 2,\,\infty } \right)$$
$$\eqalign{
& {\text{Let }}f\left( x \right) = f\left( y \right),\,x,\,y\, \in \left[ { - 2,\,\infty } \right) \cr
& \Rightarrow \left| {x + 2} \right| = \left| {y + 2} \right| \cr
& \Rightarrow x + 2 = y + 2 \cr
& \Rightarrow x = y \cr} $$
So, $$f$$ is an injective.