#### Need solution for RD Sharma maths class 12 chapter Functions exercise 2.1 question 5 sub question (x)

Surjective but not injective

Given:

$f:R \rightarrow R$, defined by $f(x)=x^3-x$

Hint:

Injective function means every element in the domain has a distinct image in the  co-domain.

surjective function means every element in the co-domain has at least one  pre image in the domain of function.

Solution:

Injective test:

Let x  and y be any two elements in the domain (R), such that

$f(x)=f(y)$

$x^3-x=y^3-y$

Here we cannot say $x=y$

For example $x=1, y=1$

$x^3-x=1-1=0$

$y^3-y=(-1)^3-(-1)= -1+1=0$

So, -1  and 1 have the same image 0.

Thus f is not an injection.

Surjection test:

Let x and y be any two elements in the domain (R) such that$f(x)=f(y)$ for some element (R) in R.

$f(x)=f(y)$

$x^3-x=y$

In order to identity surjection,

\begin{aligned} &f(x)=x^{3}-x \\ &f^{\prime}(x)=3 x^{2}-1=0 \\ &x=\pm \frac{1}{\sqrt{3}} \end{aligned}

Therefore, f is a surjective.

Therefore, f  is not a bijective.