Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions f\left(x\right)= ^ or f\left(x\right)=|x| will result in the original graph. We say that these types of graphs are symmetric about the y-axis. Figure 3-11: (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function. We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function,

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, \text f\left(x\right)= ^ \text is neither even nor odd. Also, the only function that is both even and odd is the constant function \text f\left(x\right)=0. A function is called an even function if for every input \text x f\left(x\right)=f\left(-x\right) The graph of an even function is symmetric about the y\text axis.

A function is called an odd function if for every input \text x f\left(x\right)=-f\left(-x\right) The graph of an odd function is symmetric about the origin. Given the formula for a function, determine if the function is even, odd, or neither.

Determine whether the function satisfies \text f\left(x\right)=f\left(-x\right)\text, If it does, it is even. Determine whether the function satisfies \text f\left(x\right)=-f\left(-x\right)\text, If it does, it is odd. If the function does not satisfy either rule, it is neither even nor odd.

1. Is the function \text f\left(x\right)= ^ +2x\text even, odd, or neither?

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## How do you know when a function is even or odd?

To know whether or not a function is even or odd, compare f(-x) with f(x). If f(-x) = f(x), f is even; if f(-x) = -f(x), it is odd.

## What is difference between odd and even?

Definition of Odd Numbers and Even Numbers – Odd numbers: Any natural number that can be expressed in the form of (2n + 1) is called an odd number, where n is any natural number, i.e. positive integer. Even numbers: Any natural number that can be expressed in the form of 2n is called an even number, where n is any natural number, i.e.

## What defines an odd function?

A function is odd if −f(x) = f(−x), for all x. The graph of an odd function will be symmetrical about the origin. For example, f(x) = x 3 is odd. That is, the function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.

### What is an example of an odd function?

What is an Odd Function? –

- Assume f to be a real-valued function of a variable that is real. The function f is odd when the equation is valid for all the values of x in a way that x and – x is present in the domain of the function f,
- -f(x) = f(-x)
- Or equivalently,
- f(x) + f(-x) = 0
- For example, f(x) = x 3 is an odd function, because for all value of x, -f(x) = f(-x).

#### Is 18 odd or even?

Even and Odd Numbers Even numbers can be divided evenly into groups of two. The number four can be divided into two groups of two. Odd numbers can NOT be divided evenly into groups of two. The number five can be divided into two groups of two and one group of one.

1 Odd | 2 Even | 3 Odd | 4 Even | 5 Odd | 6 Even | 7 Odd | 8 Even | 9 Odd | 10 Even | 11 Odd | 12 Even |

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1 Odd 3 Odd 5 Odd 7 Odd 9 Odd 11 Odd 2 Even 4 Even 6 Even 8 Even 10 Even 12 EvenEven and Odd Numbers

#### Is 12 odd or even?

12 is an even number.

## What four basic functions are odd?

Odd Functions: The identity function, the cubing function, the reciprocal function, the sine function.

## Can functions be both even and odd?

Yes. The constant function f(x)=0 satisfies both conditions. Hint f is even and odd ⟺f(x)=f(−x)=−f(x)⇒2f(x)=0.

#### Why is an absolute function even?

Absolute value function Since a real number and its opposite have the same absolute value, it is an even function, and is hence not invertible.