Let f : R R be a function such that f ( x ) = x 2 + 2 x + 1 x 2 + 1 . Then [2023]

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Q.
Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

1 $f (x)$ is many-one in $(- \infty, - 1)$

2 $f (x)$ is one-one in $(- \infty, \infty)$

3 $f (x)$ is many-one in $(1, \infty)$

4 $f (x)$ is one-one in $[1, \infty)$ but not in $(- \infty, \infty)$

Ans.
(4)

Given, $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1}$

So, $f (x) = 1 + \frac{2 x}{x^{2} + 1}$

$f' (x) = \frac{(x^{2} + 1) \cdot 2 - 2 x \times 2 x}{{(x^{2} + 1)}^{2}}$

$= \frac{2 - 2 x^{2}}{{(x^{2} + 1)}^{2}} = \frac{2 (1 - x^{2})}{{(x^{2} + 1)}^{2}}$

The graph of the function is, By horizontal line test, we can say that

$f (x) is one-one in [1, \infty) but not in (- \infty, \infty) .$

Solution

Q.
Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

1 $f (x)$ is many-one in $(- \infty, - 1)$

2 $f (x)$ is one-one in $(- \infty, \infty)$

3 $f (x)$ is many-one in $(1, \infty)$

4 $f (x)$ is one-one in $[1, \infty)$ but not in $(- \infty, \infty)$

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Solution

Q. Let f:R→R be a function such that f(x)=x2+2x+1x2+1. Then [2023]

1 f(x) is many-one in (-∞,-1)

2 f(x) is one-one in (-∞,∞)

3 f(x) is many-one in (1,∞)

4 f(x) is one-one in [1,∞) but not in (-∞,∞)

Ans. (4) Given, f(x)=x2+2x+1x2+1 So, f(x)=1+2xx2+1 f'(x)=(x2+1)·2-2x×2x(x2+1)2 =2-2x2(x2+1)2=2(1-x2)(x2+1)2 The graph of the function is, By horizontal line test, we can say that f(x) is one-one in [1,∞) but not in (-∞,∞).

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Q.
Let $f : R \to R$ be a function such that $f (x) = \frac{x^{2} + 2 x + 1}{x^{2} + 1} .$ Then [2023]

1 $f (x)$ is many-one in $(- \infty, - 1)$

2 $f (x)$ is one-one in $(- \infty, \infty)$

3 $f (x)$ is many-one in $(1, \infty)$

4 $f (x)$ is one-one in $[1, \infty)$ but not in $(- \infty, \infty)$