Q. The function $f:R \rightarrow R$ given by $f(x) = x^{3} - 1$ is
A) A one-one function
B) An onto function
C) A bijection
D) Neither one-one nor onto
Explanation
(c)
Given, $f(x) = x^{3} - 1$
Let $x_{1},\ x_{2} \in R$
Now, $f\left( x_{1} \right) = f\left( x_{2} \right)$
$\Rightarrow \ \ \ x_{1}^{3} - 1 = x_{2}^{3} - 1$
$\Rightarrow \ \ \ x_{1}^{3} = x_{2}^{3}$
$\Rightarrow \ \ \ \ x_{1} = x_{2}$
$\therefore f(x)$ is one-one. Also, it is onto as range of $f = R$
Hence, it is a bijection.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.