Step: 1

Function even means f (-x ) = f (x ).

Step: 2

It is possible the curve has symmetry with respect to y -axis.

Correct Answer is : has symmetry with respect to y -axis

Step: 1

[Formula.]

Step: 2

Step: 3

= e - 6 t - e 6 t 2

Step: 4

= - [( e 6 t - e - 6 t ) 2 ]

Step: 5

= - f (t )

Step: 6

Since f (- t ) = - f (t ), the given function is an odd function.

Correct Answer is : An odd function

Step: 1

[Since, cosh (x ) = e x + e - x 2 .]

Step: 2

Step: 3

= ( e 6 t + e - 6 t ) 2 = f (t )

Step: 4

Since f (- t ) = f (t ), the given function is an even function.

Correct Answer is : An even function

I.

II.

III.

IV.

V.

Step: 1

Step: 2

Step: 3

Step: 4

[If f (- x ) = - f (x ), then f (x ) is an odd function.]

Step: 5

So, f (x ) is symmetric about origin.

Correct Answer is : I and II

Step: 1

If x is the measure of an angle with terminal side in quadrant I, then - x is the measure of an angle with terminal side in quadrant IV. Similarly , if x is the measure of a quadrant IV angle, then - x is the measure of a quadrant I angle. In either case, since csc x is negative in quadrant IV and positive in quadrant I, then csc(- x ) = - csc x .

Step: 2

If x is the measure of an angle with terminal side in quadrant II, then - x is the measure of an angle with terminal side in quadrant III. Also, if x is the measure of a quadrant III angle, then - x is the measure of a quadrant II angle. In either case, since csc x is negative in quadrant III and positive in quadrant II. So, csc (- x ) = - csc x

Step: 3

Since csc(- x ) = - csc x for all values of x in the domain of the function, y = csc x is an odd function.

Correct Answer is : odd function

Step: 1

[Given function.]

Step: 2

A function, f is an even function, if f (- x ) = f (x )

Step: 3

Step: 4

A function, f is an odd function, if f (- x ) = - f (x )

Step: 5

- g (- x ) = 2x ^{3} - x + 3

Step: 6

Step: 7

Thus the function g (x ) is **neither even nor odd.**

Correct Answer is : neither even nor odd

Step: 1

Step: 2

Step: 3

Since both f (x ) and f (- x ) are equal, the function is an ** even function**.

Correct Answer is : even function

Step: 1

Step: 2

Step: 3

Since f (- x ) = - f (x ), the function is an ** odd function.**

Correct Answer is : odd function

Step: 1

Step: 2

A function, f is an even function, if f (- x ) = f (x )

Step: 3

Step: 4

A function, f is an odd function, if f (- x ) = - f (x )

Step: 5

- f (- x ) = - e - 7 x

Step: 6

Since f (x ) ≠ f (- x ) or f (x ) ≠ - f (- x ), the function f is **neither even nor odd.**

Correct Answer is : neither even nor odd

Step: 1

Step: 2

Step: 3

Since both g (x ) and g (- x ) are equal, the function is an **even function.**

Correct Answer is : even function

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