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Please solve RD Sharma class 12 Chapter Functions exercise 2.1 question 1 sub question (i) maths textbook solution.

Answers (1)

$f(x)=2x$

Given:

The example for a function which is one-one but not onto.

Hint:

One-one function means every element in the domain has a distinct image in the co-domain

. $f(x_1)=f(x_2)$ , then $x_1=x_2$

Where, $X_1,X_2$ domain of $f(x)$

Onto function defined as every element in the co-domain has at least one pre image in the domain of function.

Solution:

Here, we need to give an example of a function for one-one but not onto.

Let we consider the function,

$f:N \rightarrow N$ , given by $f(x)=2N$

Let us consider two elements $x_1$ and $x_2$ in the domain of f, so we get

$f(x_1)=2x_1$ and $f(x_2)=x_2$

Now we know the condition for finding a one-one function.

$2x_1= 2x_2$

$x_1= x_2$

Hence, the function is one-one.

Now, we need to prove $f(x)=2x$ is not onto

Let $f(x)=y$ , such that x, $y\in N$

We get,

$2x=y$

$x=\frac{y}{2}$

If we put $y=1,$

$x=\frac{1}{2}=0.5$ , which is cannot be true as $x \in N$ supposed in solution.

Hence, the given function is not onto.

So, $f(x)=2x$ is an example of one-one but not onto function.

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