The function f-R- 1-e defined by fx-X2-e-X2-1 is

F(x)=x^2+e/x^2+1 f^'(x)=2x(x^2+1) 2x(x^2+e)/(x^2+1)^2=2x^3+2x 2x^3 2ex/(x^2+1)^2 =2x 2xe/(x^2+1)^2=2x(1 e)/(x^2+1)^2 lt;0 f^'(x) lt;0, f(x) is decreasin ...

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Standard XII

Mathematics

Definition of Functions

The function ...

Question

$The function$ $f : R + \to (1, e) defined by f (x) = \frac{X^{2} + e}{X^{2} + 1}$ is

One-one but not onto

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Onto but not one-one

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Both one-one and onto

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Neither one-one nor onto

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Solution

The correct option is C
Both one-one and onto

$f (x) = \frac{x^{2} + e}{x^{2} + 1} f^{^{'}} (x) = \frac{2 x (x^{2} + 1) - 2 x (x^{2} + e)}{(x^{2} + 1)^{2}} = \frac{2 x^{3} + 2 x - 2 x^{3} - 2 e x}{(x^{2} + 1)^{2}} = \frac{2 x - 2 x e}{(x^{2} + 1)^{2}} = \frac{2 x (1 - e)}{(x^{2} + 1)^{2}} < 0 f^{^{'}} (x) < 0, f (x) is decreasing$ $Hence f is one-one function .$ $x \to 0, f (x) \to e$ $x \to \infty, f (x) \to e$ $Hence range = (1, e) = co-domain$

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Similar questions

f : R \to R

is defined by

f (x) = \frac{e^{x^{2}} - e^{{- x}^{2}}}{e^{x^{2}} + e^{{- x}^{2}}}

Q. Let

f : R \to R

defined by

f (x) = \frac{e^{x^{2}} - e^{- x^{2}}}{e^{x^{2}} + e^{{- x}^{2}}},

then

Q. The function

f : R + \to (1, e) d e f i n e d b y f (x) = \frac{X^{2} + e}{X^{2} + 1}

Q. Let

f : R \to R

be defined by

f (x) = ln (x + \sqrt{x^{2} + 1})

, then number of solution of

| f^{- 1} x | = e^{| x |}

f : R \to [- 1, 1]

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

Find area by

y = f (x)

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The function f:R+→(1,e) defined by f(x)=X2+eX2+1 is

The correct option is C Both one-one and onto f(x)=x2+ex2+1f′(x)=2x(x2+1)−2x(x2+e)(x2+1)2=2x3+2x−2x3−2ex(x2+1)2=2x−2xe(x2+1)2=2x(1−e)(x2+1)2<0f′(x)<0,f(x) is decreasing Hence f is one-one function. x→0,f(x)→e x→∞,f(x)→e Hence range = (1, e) = co-domain

$The function$ $f : R + \to (1, e) defined by f (x) = \frac{X^{2} + e}{X^{2} + 1}$ is