Question

Consider the following statements :
Statement 1 : The function $f:R\to R$ such that $f\left(x\right)=x^3$ for all $x\in R$ is one-one.
Statement 2 : $f\left(a\right)=f\left(b\right)\Rightarrow a=b$ for all $a,b\in R$ if the function $f$ is one-one.
Which one of the following is correct in respect of the above statements?

• A. Both the statements are true and Statement 2 is the correct explanation of Statement 1.
• B. Both the statements are true and Statement 2 is not the correct explanation of Statement 1.
• C. Statement 1 is true but Statement 2 is false.
• D. Statement 1 is true but Statement 2 is true.

Answer 1

Answer: (A)

Both the statements are true and Statement 2 is the correct explanation of Statement 1.

Solution:

Statement $1$ is correct, since $f\left(x\right)=x^3$

For any $x,y\in R$

$f\left(x\right)=f\left(y\right)$

$\Rightarrow x^3=y^3$

$\Rightarrow x=y$

$\therefore f$ is one-one $\forall x\in R.$

$\because f\left(a\right)=f\left(b\right)⟹a^3=b^3⟹a=b$

Statement $2$ is also correct and is the correct explanation of statement $1.$

Hence, A is the correct option.