Question

Let R be the set of real numbers. If $f:R\to R$ is a function defined by $f\left(x\right)=x^2$, then $f$ is

• A. injective but not surjective
• B. surjective but not injective
• C. bijiective
• D. none of these

Answer 1

Answer: (D)

none of these

$f\left(x_1\right)=f\left(x_2\right)$

$\Rightarrow$$(x_1{)}^2=(x_2{)}^2$

$\Rightarrow$$(x_1{)}^2-(x_2{)}^2=0$

$\Rightarrow$$(x_1-x_2)(x_1+x_2)=0$

$x_1=x_2$ & $x_1=-x_2$

Hence $f\left(x\right)$ is a many one function

The domain of $f\left(x\right)$ is $R$ while its range is $R^{+}$

Hence, option 'D' is correct.