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

**Expert Answer**

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

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

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

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

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

**Even functions** – If 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** about the $ y -axis $. Thus, reflections across the $ y-axis $ do not affect the **appearance of the function**. **Good examples of even functions** include: (integer $ n $); $\ cos ( x ) $ , $ \ cos h( x ) $ and $ | x | $.

**Odd Functions** – If given function sayy $ f ( x ) $ is an **odd function**, then for every $ x $ and $ − x $ in the** domain** 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** of the function. **Good examples of odd functions** include: (integer $ n $); $ \sin ( x )$ and $ \sin h ( x ) $.

**Numerical Result**

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

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

\[ sin (-x ) = – sin( x ) \]

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