Question

Let $f:R\to R$ where $f\left(x\right)=x^2-x-2$ then $f\left(x\right)$ is

• A. one-one and onto
• B. onto but not on-one
• C. niether one-one nor onto
• D. one-one but not onto

Answer 1

Answer: (D)

one-one but not onto

$\begin{array}{c}f\left(x\right)=x^2-x-2\\ f\left(x_1\right)=f\left(x_2\right)\\ \Rightarrow {x_1}^2-x_1-2={x_2}^2-x_2-2\\ \Rightarrow \left({x_1}^2-{x_2}^2\right)-x_1+x_2=0\\ \Rightarrow \left(x_1+x_2\right)\left(x_1-x_2\right)-1\left(x_1-x_2\right)=0\\ \Rightarrow \left(x_1+x_2-1\right)\left(x_1-x_2\right)=0\\ x_1=x_2\end{array}$
$f\left(x\right)$ is one-one
$\begin{array}{c}y=x^2-x-2\\ x^2-x-2-y=0\\ x^2-x-\left(2+y\right)=0\\ x=\frac{1\pm \sqrt{1+4\left(2+y\right)}}{2}=\frac{1\pm \sqrt{1+8+y}}{2}\\ 9+y\ge 0\Rightarrow y\ge -9\end{array}$
Co-domain $\ne$ Range
Function is not onto