**Even and odd functions :**

Let f(x) be a function.

To find whether f(x) is even or odd, we have to replace "x" by "-x" in f(x)

Now, let us plug x = -x in f(x) and find whether f(x) is odd or even.

**f(-x) = f(x) -----> f(x) is even function**

**f(-x) = - f(x) -----> f(x) is odd function**

Even though we have many applications of even and odd functions, let us consider the important application of even and odd functions in integral calculus.

**If f(x) is even function, **

**If f(x) is odd function, **

**Problem 1 : **

Let f(x) = x³, 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) = -x³

f(-x) = - f(x)

**Hence, f(x) is odd function**

**Problem 2 : **

Let f(x) = x² + 2, 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)² + 2

f(-x) = x² + 2

f(-x) = f(x)

**Hence, f(x) is even function.**

**Problem 3 : **

Let f(x) = x³ + 5, 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 + 5

f(-x) = -x³ + 5

Here we can not define f(-x) in terms of f(x).

**Hence, f(x) is neither odd nor 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 4 : **

Is sinx odd or even function ?

**Solution : **

Let f(x) = sinx

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

Then, we have

f(-x) = sin(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "sin" is negative.

So, we have

f(-x) = - sinx

f(-x) = - f(x)

f(x) is odd function

**Hence, sinx is odd function.**

**Problem 5 : **

Is cscx odd or even function ?

**Solution : **

Let f(x) = cscx

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

Then, we have

f(-x) = csc(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "csc" is negative.

So, we have

f(-x) = - cscx

f(-x) = - f(x)

f(x) is odd function

**Hence, cscx is odd function.**

**Problem 6 : **

Is secx odd or even function ?

**Solution : **

Let f(x) = secx

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

Then, we have

f(-x) = sec(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "sec" is positive.

So, we have

f(-x) = secx

f(-x) = f(x)

f(x) is even function

**Hence, secx is even function.**

**Problem 7 :**** **

Is cosx 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) = cos(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "cos" is positive.

So, we have

f(-x) = cosx

f(-x) = f(x)

f(x) is even function

**H****ence, cosx is even function.**

**Problem 8 :**** **

Is tanx odd or even function ?

**Solution : **

Let f(x) = tanx

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

Then, we have

f(-x) = tan(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant "cos" is negative.

So, we have

f(-x) = - tanx

f(-x) = - f(x)

f(x) is odd function

**Hence, tanx is odd function.**

**Problem 9 :**** **

Let f(x) = sinx + tanx, 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) = sin (-x) + tan(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant both "sin" and "tan" are negative.

So, we have

f(-x) = - sinx - tanx

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

f(-x) = - f(x)

**H****ence, f(x) is odd function.**

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

**Problem 10 :**** **

Let f(x) = secx + cosx, 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) = sec(-x) + cos(-x)

Since the angle is negative, it falls in the IV th quadrant. In IVth quadrant both "sec" and "cos" are positive.

So, we have

f(-x) = secx + cosx

f(-x) = f(x)

**H****ence, f(x) is even function.**

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

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