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). We have to conclude f(x) as even or odd function from the result of f(-x) as shown below.
1. If f(-x) = f(x), then f(x) is even function
2. If f(-x) = -f(x), then f(x) is odd function
If f(-x) is neither equal to f(x) nor -f(x), we have to conclude f(x) is neither even nor odd.
The sum or difference of two even functions is always even.
The sum or difference of two odd functions is always odd.
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,
Example 1 :
Let f(x) = x^{3}, 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}
f(-x) = (-x)^{3}
= -x^{3}
= -f(x)
Since f(-x) = -f(x),
f(x) is an odd function
Example 2 :
Let f(x) = x^{2} + 2, is f(x) odd or even function?
Solution :
f(x) = x^{2} + 2
f(-x) = (-x)^{2} + 2
= x^{2} + 2
= f(x)
Since f(-x) = f(x),
f(x) is an even function
Example 3 :
Let f(x) = x^{3} - 2x, is f(x) odd or even function?
Solution :
f(x) = x^{3} - 2x
f(-x) = (-x)^{3} - 2(-x)
= -x^{3} + 2x
= -(x^{3} - 2x)
= -f(x)
Since f(-x) = -f(x),
f(x) is an odd function
Example 4 :
Let f(x) = 5x^{3} + x^{2 }- 1, is f(x) odd or even function?
Solution :
f(x) = 5x^{3} + x^{2 }- 1
f(-x) = 5(-x)^{3} + (-x)^{2 }- 1
= 5(-x^{3}) - x^{2 }- 1
= -5x^{3} - x^{2 }- 1
= -(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.
Example 5 :
Let f(x) = x^{4} + 2x^{2} + 5, is f(x) odd or even function?
Solution :
f(x) = x^{4} + 2x^{2} + 5
f(-x) = (-x)^{4 }+ 2(-x)^{2} + 2
= x^{4 }+ 2x^{2} + 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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