How to Find Even and Odd Functions?

Even function

For a real-valued function \(f(x)\), when the output value of \(f(-x)\) is the same as \(f(x)\), for all values of \(x\) in the domain of \(f\), the function is an even function. An even function must have the following equation:

\(\color{blue}{f(-x) = f(x)}\)

For all values of \(x\) in \(D\: (f)\), where \(D\: (f)\) is the domain of \(f\).

 In other words, we can say that the equation \(f(-x)\: – f(x) = 0\) holds for an even function, for all \(x\).

Odd function

For a real-valued function \(f(x)\), when the output value of \(f(-x)\) is the same as the negative of \(f(x)\), for all values of \(x\) in the domain of \(f\), the function is an odd function. An odd function should hold the following equation:

\(\color{blue}{f(-x) = -f(x)}\)

For all values of \(x\) in \(D\:(f)\), where \(D\:(f)\) is the domain of the function \(f\).

In other words, we can say that the equation \(f(-x) + f(x) = 0\) holds for an odd function, for all \(x\).

Neither even nor odd function

A real-valued function \(f(x)\) is neither even nor odd if it does not satisfy \(\color{blue}{f(-x) = f(x)}\) and \(\color{blue}{f(-x) = -f(x)}\) for at least one value of \(x\) in the domain of the function \(f(x)\).

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Both even and odd functions

A real-valued function \(f(x)\) is said to be both even and odd if:

\(\color{blue}{f(-x) = f(x)}\) and \(\color{blue}{f(-x) = -f(x)}\)

For all values of \(x\) in the domain of the function \(f(x)\). There is only one function that is both even and odd and that is the zero function, \(f(x) = 0\) for all \(x\).

Properties of even and odd functions

• The sum of two odd functions is odd and the sum of two even functions is even.
• The difference between two odd functions is odd and the difference between two even functions is even.
• The sum of an odd and even function is neither even nor odd unless one of them is a zero function.
• The product of two odd functions is also an even function and the product of two even functions is even.
• The product of an even and an odd function is odd.
• The quotient of two odd functions is an even function and the quotient of two even functions is even.
• The quotient of an even and an odd function is odd.
• The composition of two odd functions is odd and the composition of two even functions is even.
• The composition of an even and an odd function is even.

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Even and odd functions – Example 1:

Identify whether the following function is even, odd, or neither. \(f(x)= 5x^4+4x^2+2\)

Solution: For this, it is enough to put \(-x\) in the equation of the function and simplify:

\(f(x)= 5x^4+4x^2+2\)

\(= 5(-x^4)+4(-x^2)+2\)

\(f(x)= 5x^4+4x^2+2\)

\(f(-x)= f(x)\)→ This means that the function is even.

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