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

$f(x)=2x$

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.