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What is the difference between f(-x) and -f(x)?

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.

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