# How to Find Even and Odd Functions

There are different types of functions in mathematics. We can determine whether a function is odd or even algebraically and graphically. In this step-by-step guide, you will learn more about even and odd functions and how to solve them.

## Step-by-step guide toeven and odd functions

There are different types of functions in mathematics that we study. We can determine whether a function is even or odd algebraically and graphically.

### 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 $$x$$.

### 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 $$x$$.

### 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 $$f(x)$$.

### 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.

### Even and odd functions graph

The graph of an even function is symmetric about the $$y$$-axis. In other words, the graph of an even function remains constant after reflection on the $$y$$-axis. While the graph of an odd function is symmetric about the origin $$(0, 0)$$. In other words, the graph of an odd function is at the same distance from the origin but in opposite directions.

### 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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