Question

The function $$f\left( x \right) = \log \left( {x + \sqrt {{x^2} + 1} } \right),$$      is

A. neither an even nor an odd function
B. an even function
C. an odd function  
D. a periodic function.
Answer :   an odd function
Solution :
$$\eqalign{ & f\left( x \right) = \log \left( {x + \sqrt {{x^2} + 1} } \right) \cr & f\left( { - x} \right) = \log \left\{ { - x + \sqrt {{x^2} + 1} } \right\} = \log \left\{ {\frac{{ - {x^2} + {x^2} + 1}}{{x + \sqrt {{x^2} + 1} }}} \right\} \cr & = - \log \left( {x + \sqrt {{x^2} + 1} } \right) = - f\left( x \right) \cr & \Rightarrow f\left( x \right){\text{ is an odd function}}{\text{.}} \cr} $$

Releted MCQ Question on
Calculus >> Function

Releted Question 1

Let $$R$$ be the set of real numbers. If $$f:R \to R$$   is a function defined by $$f\left( x \right) = {x^2},$$   then $$f$$ is:

A. Injective but not surjective
B. Surjective but not injective
C. Bijective
D. None of these.
View Answer
Releted Question 2

The entire graphs of the equation $$y = {x^2} + kx - x + 9$$     is strictly above the $$x$$-axis if and only if

A. $$k < 7$$
B. $$ - 5 < k < 7$$
C. $$k > - 5$$
D. None of these.
View Answer
Releted Question 3

Let $$f\left( x \right) = \left| {x - 1} \right|.$$    Then

A. $$f\left( {{x^2}} \right) = {\left( {f\left( x \right)} \right)^2}$$
B. $$f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$
C. $$f\left( {\left| x \right|} \right) = \left| {f\left( x \right)} \right|$$
D. None of these
View Answer
Releted Question 4

If $$x$$ satisfies $$\left| {x - 1} \right| + \left| {x - 2} \right| + \left| {x - 3} \right| \geqslant 6,$$       then

A. $$0 \leqslant x \leqslant 4$$
B. $$x \leqslant - 2\,{\text{or}}\,x \geqslant 4$$
C. $$x \leqslant 0\,{\text{or}}\,x \geqslant 4$$
D. None of these
View Answer

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