Let a function f - R - R be defined as f x - x 2- x 2-1- x 2 for all x - R- Then state whether f is one-one or onto -

F(x)=x^2 x^21+x^2 ⇒ f(x)=x^41+x^2 Now, replace x by x, f( x)=( x)^41+( x)^2=x^41+x^2 ⇒ f( x)=f(x) nbsp; ⇒ f(x) is an even function, hence it is many one. Si ...

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Standard XII

Mathematics

Convexity

Let a functio...

Question

Let a function $f : R \to R$ be defined as $f (x) = x^{2} - \frac{x^{2}}{1 + x^{2}}$ for all $x \in R$ . Then state whether $f$ is one-one or onto ?

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Solution

$f (x) = x^{2} - \frac{x^{2}}{1 + x^{2}}$
$\Rightarrow f (x) = \frac{x^{4}}{1 + x^{2}}$
Now, replace $x$ by $- x$ ,
$f (- x) = \frac{(- x)^{4}}{1 + (- x)^{2}} = \frac{x^{4}}{1 + x^{2}}$
$\Rightarrow f (- x) = f (x)$
$\Rightarrow f (x)$ is an even function, hence it is many one.

Since, $f (x) \geq 0 \forall x \in R$ , so $f (x)$ does not take any negative value.
So, range of $f (x)$ is not $R$ .
Hence, it is not onto.
$\Rightarrow f (x)$ is neither one-one nor onto.

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Let a function f:R→R be defined as f(x)=x2−x21+x2 for all x∈R. Then state whether f is one-one or onto ?

Let a function $f : R \to R$ be defined as $f (x) = x^{2} - \frac{x^{2}}{1 + x^{2}}$ for all $x \in R$ . Then state whether $f$ is one-one or onto ?