The main objective of this question is to check whether the **addition** of the given two functions when **both the functions** are **odd**, **even** or **one** is **odd** and the other is** even**Â results in **even or odd function**.

This question shows the concept of** even and odd functions**. AnÂ **even function** is** mathematically represented** as:

\[f(-x) = f(x)\]

While the **odd function** is **mathematically** represented as:

\[f(-x) = -f(x)\]

## Expert Answer

We have to** show** that the **given two functions**Â which areÂ $ f $ and $ g$Â are **even or odd.**

**Let**:

\[h(x) \space = \space f(x) \space + \space g(x) \]

An** even** function is **mathematically represented** as $ f(-x) \space = \space f(x) $ while the **odd function** is **mathematically** represented $ f(-x) \space = \space -f(x) $.

Suppose that the** given two functions** which are $ f $Â and $ g$ are **even functions,** then:

\[h(-x) \space = \space f(-x) \space + \space g(-x) \]

\[h(x) \space = \space f(x) \space + \space g(x) \]

**Thus,** $ h $ is an **even function**.

Now suppose that the given **two functions** which are $ f $Â and $ g$ are **odd functions,** then:

\[h(-x) \space = \space f(-x) \space + \space g(-x) \]

\[ = \space – f(x) \space + \space -g(x) \]

\[ = -( f(x) \space + \space g(x) )\]

\[ -h(x) \space =Â \spaceÂ – ( f(x) \space + \space g(x) )\]

**Thus** $ h $ is an odd function.

Now from the **given two functions**, one function is **odd** and the other is **even,** so:

\[h(-x) \space = \space f(-x) \space + \space g(-x) \]

\[h(-x) \space = \space f(x) \space + \space g(-x) \]

\[h(-x) \space = \space f(x) \space – \space g(-x) \]

This $ h$ function is neither **even nor odd**.

## Numerical Answer

- When the
**two functions are odd,**then the sum of two functions results in an**odd function**. - When the
**two functions are even,**then the sum of two functions results in an**even function**. - When
**two functions**are given; one is**odd**and the other is**even,**then their sum will result in**neither an even nor odd function**.

## Example

When the** two functions** $ aÂ $ and $ b $ are **even,**Â then the production of these two function will result in **even or odd function**.

We know that an **even function** is **mathematically** represented as:

\[f(-x) = f(x)\]

While the** odd function** is **mathematically** represented as:

\[f(-x) = -f(x)\]

**So**,Let**:**

\[f \space : \space A \space \rightarrowÂ \space f(x)\]

This is an **even function** then:

\[f(-x) \space = \space f(x)\]

**Also**, let $

\[g \space : \space B \space \rightarrowÂ \space f(x)\]

This is an** even function** then:

\[g(-x) \space = \space g(x) \]

**Let**:

\[h \space = \space h . g \]

\[h(-x) \space = \space (f.g)(-x)Â \space = \space f(-x)g(-x) \space = \space f(x)g(x) \space = \space h(x)\]

Thus, when the **two given functions** are **even** then their **product** will also **result** in an **even function**.