Question

Let $g(x)=x^2-4x-5$, then

• A. $g$ is one-one on $R$
• B. $g$ is not one-one on $R$
• C. $g$ is bijective on $R$
• D. None of these

Answer 1

Answer: (B)

Let $g(x_1)=g(x_2)$
$\Rightarrow x_1^2-4x_1-5=x_2^2-4x_2-5$
$\Rightarrow x_1^2-x_2^2=4\left(x_1-x_2\right)$
$\Rightarrow \left(x_1-x_2\right)\left(x_1+x_2-4\right)=0$
Either $x_1=x_2$ or $x_1+x_2=4$
Either $x_1=x_2$ or $x_1=4-x_2$
$\therefore$ There are two values of $x_1$, for which $g(x_1)=g(x_2)$.
$\therefore g(x)$ is not one-one $\forall x\in R$