Let f - R - R be a function defined as f x -x2-2 x -5- x 2- x -1- Then f isA- Many-one and ontoB- One-one and intoC- One-one and ontoD- Many-one and into

The correct option is D Many one and intox^2 + x + 1 gt; 0, x^2 + 2x + 5 gt; 0 ⇒ f(x) gt; 0 ⇒ f is into as range is not R f'(x)=(x^2+x+1)(2x+2) (x^2+ ...

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Standard XII

Mathematics

Graph of Quadratic Expression

Let f : R → R...

Question

Let $f : R \to R$ be a function defined as $f (x) = \frac{x^{2} + 2 x + 5}{x^{2} + x + 1}$ . Then f is

One-one and into

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One-one and onto

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Many-one and into

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Many-one and onto

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Solution

The correct option is C Many-one and into
$x^{2} + x + 1 > 0, x^{2} + 2 x + 5 > 0 \Rightarrow f (x) > 0$
$\Rightarrow$ f is into as range is not R
$f^{'} (x) = {\frac{(x^{2} + x + 1) (2 x + 2) - (x^{2} + 2 x + 5) (2 x + 1)}{(x^{2} + x + 1)}}^{2}$
f'(x) = 0 has real roots so f is many-one

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Let f:R→R be a function defined as f(x)=x2+2x+5x2+x+1. Then f is

The correct option is C Many-one and intox2+x+1>0,x2+2x+5>0⇒f(x)>0 ⇒ f is into as range is not R f′(x)=(x2+x+1)(2x+2)−(x2+2x+5)(2x+1)(x2+x+1)2 f'(x) = 0 has real roots so f is many-one

Let $f : R \to R$ be a function defined as $f (x) = \frac{x^{2} + 2 x + 5}{x^{2} + x + 1}$ . Then f is

The correct option is C Many-one and into
$x^{2} + x + 1 > 0, x^{2} + 2 x + 5 > 0 \Rightarrow f (x) > 0$
$\Rightarrow$ f is into as range is not R
$f^{'} (x) = {\frac{(x^{2} + x + 1) (2 x + 2) - (x^{2} + 2 x + 5) (2 x + 1)}{(x^{2} + x + 1)}}^{2}$
f'(x) = 0 has real roots so f is many-one