A Function can be classified as Even, Odd or Neither. This classification can be determined graphically or algebraically.
How to check if the graph is odd ?
The graph will be symmetric with respect to the origin.
In other words :
If you spin the picture upside down about the Origin, the graph looks the same!
How to check if the graph is even ?
The graph will be symmetric with respect to the y-axis.
Properties of odd function :
Properties of even function :
A function f is even when, for each x in the domain of f,
f(-x) = f(x).
A function f is odd when, for each x in the domain of f,
f(-x) = -f(x).
Determine graphically using possible symmetry, whether the following functions are even, odd or neither.
Problem 1 :
Solution :
The graph is not symmetric about origin, then it is not odd function.
The graph is not symmetric about y-axis, then it is not even function.
So, it is neither.
Problem 2 :
Solution :
Here y-axis is acting as a mirror. Clearly it is symmetric about y-axis. Then, it is even function.
Problem 3 :
Solution :
Here y-axis is acting as a mirror. Clearly it is symmetric about y-axis. Then, it is even function.
Problem 4 :
Solution :
The graph is symmetric about origin. So, it is odd function.
Problem 5 :
Solution :
The graph is not symmetric about origin, then it is not odd function.
The graph is not symmetric about y-axis, then it is not even function.
So, it is neither.
Problem 6 :
Solution :
The graph is not symmetric about origin, then it is not odd function.
The graph is not symmetric about y-axis, then it is not even function.
So, it is neither.
Verify algebraically whether each function is even, odd or neither!
Problem 1 :
f(x) = x^{3} - 6x
Solution:
f(x) = x^{3} - 6x
Put x = -x
f(-x) = (-x)^{3} - 6(-x)
= -x^{3} + 6x
So, it is neither.
Problem 2 :
g(x) = x^{4} - 2x^{2}
Solution:
g(x) = x^{4} - 2x^{2}
Put x = -x
g(-x) = (-x)^{4} - 2(-x)^{2}
= x^{4} - 2x^{2}
So, it is even function.
Problem 3 :
h(x) = x^{2} + 2x + 1
Solution:
h(x) = x^{2} + 2x + 1
Put x = -x
h(-x) = (-x)^{2} + 2(-x) + 1
= x^{2} - 2x + 1
h(-x) ≠ h(x)
So, it is neither.
Problem 4 :
f(x) = x^{2} + 6
Solution:
f(x) = x^{2} + 6
Put x = -x
f(-x) = (-x)^{2} + 6
= x^{2} + 6
f(-x) = f(x)
So, it is even function.
Problem 5 :
g(x) = 7x^{3} - x
Solution:
g(x) = 7x^{3} - x
Put x = -x
g(-x) = 7(-x)^{3} - (-x)
= -7x^{3} + x
g(-x) ≠ g(x)
So, it is odd function.
Problem 6 :
h(x) = x^{5} + 1
Solution:
h(x) = x^{5} + 1
Put x = -x
h(-x) = (-x)^{5} + 1
= -x^{5} + 1
h(-x) ≠ h(x)
So, it is neither.
Problem 7 :
f(x) = x√(4 - x^{2})
Solution:
f(x) = x√(4 - x^{2})
Put x = -x
f(-x) = (-x)√(4 - (-x)^{2})
= -x√(4 - x^{2})
So, it is neither.
Problem 8 :
g(x) = x^{4}√(1 + x)
Solution:
g(x) = x^{4}√(1 + x)
Put x = -x
g(-x) = (-x)^{4}√(1 + (-x))
= x^{4}√(1 - x)
So, it is neither.
Problem 9 :
h(x) = |x| - 1
Solution:
h(x) = |x| - 1
Put x = -x
h(-x) = |-x| - 1
= x - 1
So, it is even function.
Problem 10 :
Solution:
So, it is even function.
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