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

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

Answers (1)

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 thatf(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.

Posted by

infoexpert24

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