EVEN AND ODD FUNCTIONS

Example 1 :

Let f(x) = x³, 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) = x³

f(-x) = (-x)³

= -x³

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Example 2 :

Let f(x) = x²  + 2, is f(x) odd or even function?

Solution :

f(x) = x²  + 2

f(-x) = (-x)² + 2

= x² + 2

= f(x)

Since f(-x) = f(x),

f(x) is an even function

Example 3 :

Let f(x) = x³ - 2x, is f(x) odd or even function?

Solution :

f(x) = x³ - 2x

f(-x) = (-x)³ - 2(-x)

= -x³ +  2x

= -(x³ - 2x)

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Example 4 :

Let f(x) = 5x³ + x² - 1, is f(x) odd or even function?

Solution :

f(x) = 5x³ + x² - 1

f(-x) = 5(-x)³ + (-x)² - 1

= 5(-x³) - x² - 1

= -5x³ - x² - 1

= -(5x³ + x² + 1)

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

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

Example 5 :

Let f(x) = x⁴ + 2x²  + 5, is f(x) odd or even function?

Solution :

f(x) = x⁴ + 2x²  + 5

f(-x) = (-x)⁴ + 2(-x)² + 2

= x⁴ + 2x² + 2

 = f(x)

Since f(-x) = f(x),

f(x) is an 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.

To know more about trigonometric ratios in all the four quadrants, click here.

Example 6 :

Is sinx odd or even function?

Solution :

Let f(x) = sinx.

f(x) = sinx

f(-x) = sin(-x)

Because the angle is negative, it falls in the IV^th quadrant. In IV^th quadrant sine is negative.

f(-x) = -sinx

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Therefvore, sinx is an odd function.

Example 7 : 

Is cosx odd or even function?

Solution :

Let f(x) = cosx.

f(x) = cosx

f(-x) = cos(-x)

In IV^th quadrant cosine is positive.

f(-x) = cosx

= f(x)

Since f(-x) = f(x),

f(x) is an even function

Therefore, cosx is even function.

Example 8 : 

Is tanx odd or even function?

Solution :

Let f(x) = tanx.

f(x) = tanx

f(-x) = tan(-x)

In IV^th quadrant tangent is negative.

f(-x) = -tanx

f(-x) = -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

Therefore, tanx is odd function.

Example 9 : 

Is secx odd or even function?

Solution :

Let f(x) = secx.

f(x) = secx

f(-x) = sec(-x)

In IV^th quadrant secant is positive.

f(-x) = secx

f(-x) = f(x)

Since f(-x) = f(x),

f(x) is an even function

Therefore, secx is even function.

Example 10 :

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

Solution :

f(x) = sinx + tanx

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

f(-x) = -sinx - tanx

= -(sinx + tanx)

= -f(x)

Since f(-x) = -f(x),

f(x) is an odd function

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

Example 11 : 

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

Solution :

f(x) = secx + cosx

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

f(-x) = secx + cosx

f(-x) = f(x)

Since f(-x) = f(x),

f(x) is an even function

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

Example 12 : 

If the point (8, -3) is on the graph of y = f(x), name another point on the graph if 

(a) f(x) is an even function

(b) f(x) is an odd function 

Solution :

Since the point (8, -3) is on the graph of y = f(x),

x = 8, y = -3

When x = 8, the value of f(x) is -3.

f(8) = -3 ----(1)

Part (a) :

Given : f(x) is an even function.

f(-x) = f(x)

f(-8) = f(8)

f(-8) = -3 ........from (1)

Another point on the graph is (-8, -3).

Part (b) :

Given : f(x) is an odd function.

f(-x) = -f(x)

f(-8) = -f(8)

f(-8) = -(-3) ........from (1)

f(-8) = 3

Another point on the graph is (-8, 3).

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