This**Â article aims to determine**Â the difference between **two functions**Â and categorize them into either two types of functions:Â **odd and even**. This article uses **concepts of even and odd functions**Â and how to find whether the given function is**Â odd or even.**

**Expert Answer**

The graph of $ f ( – x ) $ is the **mirror image of graph**Â of $ f ( x ) $ with respect to **vertical axis.**

The graph of $ -f ( x ) $ is the **mirror image of graph**Â of $ f ( x ) $ with respect to **horizontal axis**.

The function is called**Â even**Â if $ f ( x ) = f ( – x ) $ for all $ x $.

The function is called **odd**Â if $ Â – f ( x ) = f ( – x ) $ for all $ x $.

Functions are described as **odd**, **even**, or **neither**. Mostly functions are **neither odd**Â **nor even**, but it’s good to know which ones are **even or odd**Â and how to determine the difference between both.

**Even functions**Â – If Â given function say $ f ( x ) $ is an **even function**, then for every $ x $ and $ – x $ in the domain of $ f $ , $ f ( x ) = f ( – x ) $. **Graphically**, the function is **symmetric**Â about the $ y -axis $. Thus, reflections across the $ y-axis $ do not affect the **appearance of the function**. **Good examples of even functions**Â include: (integer $ n $); $\ cos ( x ) $ , $ \ cos h( x ) $ and $ | x | $.

**Odd Functions**Â – If given function sayy $ f ( x ) $ is an **odd function**, then for every $ x $ and $ âˆ’ x $ in the**Â domain**Â of $ f $, $ – f ( x ) = f ( – x ) $. **Graphically**, this means that function is **rotationally symmetric about the origin**. That is, rotation of $ 180 ^ { \circ } $ or any multiple of $ 180 ^ { \circ } $ do not affect the **appearance**Â of the function. **Good examples of odd functions**Â include: (integer $ n $); $ \sin ( x )$ and $ \sin h ( x ) $.

**Numerical Result**

The function is called**Â even**Â if $ f ( x ) = f ( – x ) $ for all $ x $.

The function is called **odd**Â if $ – f ( x ) = f ( – x ) $ for all $ x $.

**Example**

**Determine whether the function $ \sin (x) $ is even or odd.**

**Solution**

The function is an **odd function. **The function is called**Â odd** if $ Â – f ( x ) = f ( – x ) $ for all $ x $. For $ \ sin ( x ) $

\[ sin (-x ) = – sin( x ) \]

Hence, the function $ \sin (x) $ is an **odd function.**