Determine if the following functions are even or odd:

Exercise 1

f(x) = { x }^{ 6 } + { x }^{ 4 } - { x }^{ 2 }

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Exercise 2

f(x) = { x }^{ 5 } + { x }^{ 3 } - x

Exercise 3

f(x) = x \left | x \right |

Exercise 4

f(x) = \left | x \right | - 1

Exercise 5

f(x) = \frac { { x }^{ 2 } }{ 1 - { x }^{ 2 } }

Exercise 6

f(x) = \frac { x }{ 1 - { x }^{ 2 } }

 

 

Solution of exercise 1

Determine if the function is even or odd.

f(x) = { x }^{ 6 } + { x }^{ 4 } - { x }^{ 2 }

f(-x) = { (-x) }^{ 6 } + { (-x) }^{ 4 } - { (-x) }^{ 2 } = { x }^{ 6 } + { x }^{ 4 } - { x }^{ 2 }

The function is even and symmetrical about the vertical axis.

 

Solution of exercise 2

Determine if the function is even or odd.

f(x) = { x }^{ 5 } + { x }^{ 3 } - x

f(-x) = { (-x) }^{ 5 } + { (-x) }^{ 3 } - (-x) = - { x }^{ 5 } - { x }^{ 3 } + x

The function is odd and symmetrical about the origin.

 

Solution of exercise 3

Determine if the function is even or odd.

f(x) = x \left | x \right |

f(-x) = - x \left | (-x) \right | = - x \left | x \right |

The function is odd and symmetrical about the origin.

 

Solution of exercise 4

Determine if the function is even or odd.

f(x) = \left | x \right | - 1

f(-x) = \left | (-x) \right | - 1 = \left | x \right | - 1

The function is even and symmetrical about the vertical axis.

 

Solution of exercise 5

Determine if functions are even or odd:

f(x) = \frac { { x }^{ 2 } }{ 1 - { x }^{ 2 } }

f(-x) = \frac { { (-x) }^{ 2 } }{ 1 - { (-x) }^{ 2 } } = \frac { { x }^{ 2 } }{ 1 - { x }^{ 2 } }

The function is even and symmetrical about the vertical axis.

 

Solution of exercise 6

Determine if the function is even or odd.

f(x) = \frac { x }{ 1 - { x }^{ 2 } }

f(-x) = \frac { (-x) }{ 1 - { (-x) }^{ 2 } } = \frac { - x }{ 1 - { x }^{ 2 } }

The function is odd and symmetrical about the origin.

 

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.