Mark (√) against the correct answer in the following: f : R → R : f(x) = $x^2$ is A. one - one and onto B. one - one and into C. many - one and onto D. many - one and into

Home » Mark (√) against the correct answer in the following: f : R → R : f(x) = is A. one – one and onto B. one – one and into C. many – one and onto D. many – one and into

Mark (√) against the correct answer in the following:
f : R → R : f(x) = $x^2$ is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

CBSE | Class 12 | Functions | Maths | RS Aggarwal

Mark (√) against the correct answer in the following:
f : R → R : f(x) = $x^2$ is
A. one – one and onto
B. one – one and into
C. many – one and onto
D. many – one and into

Solution:

Option(D) is correct.
One-One function
Suppose $\mathrm{p}, \mathrm{q}$ be two arbitrary elements in $\mathrm{R}$
Therefore, $f(p)=f(q)$
$\begin{array}{l} \Rightarrow \mathrm{p}_{2}=\mathrm{q}_{2} \\ \Rightarrow \mathrm{p}=\mathrm{q} \text { and }-\mathrm{q} \end{array}$
Therefore, $\mathrm{f}(\mathrm{x})$ is many-one function.
Onto function
Suppose $v$ be an arbitrary element of $R$ (Co-domain)
Therefore, $f(x)=v$
$\begin{array}{l} \mathrm{x} 2=\mathrm{v} \\ \Rightarrow \mathrm{x}=\sqrt{v} \end{array}$
As $\mathrm{v} \in \mathrm{R}$
If $\mathrm{v}=2, \sqrt{v}=1.1414$ , which is not possible as $\mathrm{x} \in \mathrm{R}$
Hence, $\mathrm{f}(\mathrm{x})$ is not onto function. It is into function.