DETERMINE GRAPHICALLY OR ALGEBRAICALLY IF A FUNCTION IS ODD OR EVEN

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!

odd-function-from-graph

How to check if the graph is even ?

The graph will be symmetric with respect to the y-axis.

even-frunction-from-graph

Properties of odd and even functions :

Properties of odd function :

  • The graph is symmetric about origin.
  • The exponents of all terms in its equation are odd.

Properties of even function :

  • The graph is symmetric about y-axis.
  • The exponents of all terms in its equation are even.

Algebraically Check if it is odd or even

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).

Graphically Checking If the Function is Odd or Even

Determine graphically using possible symmetry, whether the following functions are even, odd or neither.

Problem 1 :

even-odd-function-q1.png

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 :

even-odd-function-q2.png

Solution :

Here y-axis is acting as a mirror. Clearly it is symmetric about y-axis. Then, it is even function.

Problem 3 :

even-odd-function-q3.png

Solution :

Here y-axis is acting as a mirror. Clearly it is symmetric about y-axis. Then, it is even function.

Problem 4 :

even-odd-function-q4.png

Solution :

The graph is symmetric about origin. So, it is odd function.

Problem 5 :

even-odd-function-q5.png

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 :

even-odd-function-q6.png

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.

Problems on Algebraically Checking If It is Odd or Even

Verify algebraically whether each function is even, odd or neither!

Problem 1 :

f(x) = x3 - 6x

Solution:

f(x) = x3 - 6x

Put x = -x

f(-x) = (-x)3 - 6(-x)

= -x3 + 6x

So, it is neither.

Problem 2 :

g(x) = x4 - 2x2

Solution:

g(x) = x4 - 2x2

Put x = -x

g(-x) = (-x)4 - 2(-x)2

= x4 - 2x2

So, it is even function.

Problem 3 :

h(x) = x2 + 2x + 1

Solution:

h(x) = x2 + 2x + 1

Put x = -x

h(-x) = (-x)2 + 2(-x) + 1

= x2 - 2x + 1

h(-x) ≠ h(x)

So, it is neither.

Problem 4 :

f(x) = x2 + 6

Solution:

f(x) = x2 + 6

Put x = -x

f(-x) = (-x)2 + 6

= x2 + 6

f(-x) = f(x)

So, it is even function.

Problem 5 :

g(x) = 7x3 - x

Solution:

g(x) = 7x3 - x

Put x = -x

g(-x) = 7(-x)3 - (-x)

= -7x3 + x

g(-x) ≠ g(x)

So, it is odd function.

Problem 6 :

h(x) = x5 + 1

Solution:

h(x) = x5 + 1

Put x = -x

h(-x) = (-x)5 + 1

= -x5 + 1

h(-x) ≠ h(x)

So, it is neither.

Problem 7 :

f(x) = x√(4 - x2)

Solution:

f(x) = x√(4 - x2)

Put x = -x

f(-x) = (-x)√(4 - (-x)2)

= -x√(4 - x2)

So, it is neither.

Problem 8 :

g(x) = x4√(1 + x)

Solution:

g(x) = x4√(1 + x)

Put x = -x

g(-x) = (-x)4√(1 + (-x))

= x4√(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 :

g(x)=14x6-5x2

Solution:

g(x)=14x6-5x2Put x=-xg(-x)=14(-x)6-5(-x)2=14x6-5x2

So, it is even function.

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