**Problem 1 : **

Let f(x) = x^{3}, is f(x) odd or even function ?

**Problem 2 : **

Let f(x) = x^{2} + 2, is f(x) odd or even function ?

**Problem 3 : **

Let f(x) = x^{3} - 2x, is f(x) odd or even function ?

**Problem 4 : **

Let f(x) = 5x^{3} + x^{2 }- 1, is f(x) odd or even function ?

**Problem 5 : **

Let f(x) = x^{4} + 2x^{2} + 5, is f(x) odd or even function ?

**Problem 6 : **

Is sinx odd or even function ?

**Problem 7 : **

Is cscx odd or even function ?

**Problem 8 : **

Is secx odd or even function ?

**Problem 9 :**** **

Is cosx odd or even function ?

**Problem 10 :**** **

Is tanx odd or even function ?

**Problem 11 :**** **

Let f(x) = sinx + tanx, is f(x) odd or even function ?

**Problem 12 :**** **

Let f(x) = secx + cosx, is f(x) odd or even function ?

**Problem 1 : **

Let f(x) = x^{3}, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, let us plug x = -x in f(x).

Then, we have

f(-x) = (-x)^{3}

f(-x) = -x^{3}

f(-x) = - f(x)

So, f(x) is odd function.

**Problem 2 : **

Let f(x) = x^{2} + 2, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{2} + 2

f(-x) = x^{2} + 2

f(-x) = f(x)

So, f(x) is even function.

**Problem 3 : **

Let f(x) = x^{3} - 2x, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{3} - 2(-x)

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

f(-x) = -(x^{3} - 2x)

f(-x) = -f(x)

So, f(x) is odd function.

**Problem 4 : **

Let f(x) = 5x^{3} + x^{2 }- 1, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = 5(-x)^{3} + (-x)^{2 }- 1

f(-x) = 5(-x^{3}) - x^{2 }- 1

f(-x) = -5x^{3} - x^{2 }- 1

f(-x) = -(5x^{3} + x^{2 }+ 1)

f(-x) can not be expressed as either as f(x) or -f(x).

So, f(x) is neither even nor odd function.

**Problem 5 : **

Let f(x) = x^{4} + 2x^{2} + 5, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = (-x)^{4 }+ 2(-x)^{2} + 2

f(-x) = x^{4 }+ 2x^{2} + 2

f(-x) = f(x)

So, f(x) is even function.

**Important Note : **

In trigonometric ratios, if we have negative angle, we have to understand that the angle will fall in the IV^{th }quadrant.

In IV^{th} quadrant, the trigonometric ratios "cos" and "sec" are positive and all other trigonometric ratios are negative

**Problem 6 : **

Is sinx odd or even function ?

**Solution : **

Let f(x) = sinx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sin(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "sin" is negative.

So, we have

f(-x) = - sinx

f(-x) = - f(x)

f(x) is odd function

So, sinx is odd function.

**Problem 7 : **

Is cscx odd or even function ?

**Solution : **

Let f(x) = cscx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = csc(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "csc" is negative.

So, we have

f(-x) = - cscx

f(-x) = - f(x)

f(x) is odd function

So, cscx is odd function.

**Problem 8 : **

Is secx odd or even function ?

**Solution : **

Let f(x) = secx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sec(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "sec" is positive.

So, we have

f(-x) = secx

f(-x) = f(x)

f(x) is even function

So, secx is even function.

**Problem 9 :**** **

Is cosx odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = cos(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "cos" is positive.

So, we have

f(-x) = cosx

f(-x) = f(x)

f(x) is even function

So, cosx is even function.

**Problem 10 :**** **

Is tanx odd or even function ?

**Solution : **

Let f(x) = tanx

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = tan(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant "tan" is negative.

So, we have

f(-x) = - tanx

f(-x) = - f(x)

f(x) is odd function

So, tanx is odd function.

**Problem 11 :**** **

Let f(x) = sinx + tanx, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sin (-x) + tan(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant both "sin" and "tan" are negative.

So, we have

f(-x) = - sinx - tanx

f(-x) = - (sinx + tanx)

f(-x) = - f(x)

So, f(x) is odd function.

**Note : **The sum or difference of two odd functions is always odd.

**Problem 12 :**** **

Let f(x) = secx + cosx, is f(x) odd or even function ?

**Solution : **

To know f(x) is odd or even function, substitute -x for x in f(x).

Then, we have

f(-x) = sec(-x) + cos(-x)

Because the angle is negative, it falls in the IV^{th} quadrant. In IV^{th} quadrant both "sec" and "cos" are positive.

So, we have

f(-x) = secx + cosx

f(-x) = f(x)

So, f(x) is even function.

**Note : **The sum or difference of two even functions is always even.

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