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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Before we begin, let's recall two important definitions.

- A function $f$ has even symmetry when $f(-x)=f(x)$ for all $x$ in its domain.
- A function $f$ has odd symmetry when $f(-x)=-f(x)$ for all $x$ in its domain.

Reset

$f(-x)=(-x)_{5}+3(-x)_{3}−(-x)$

Simplify right-hand side

NegBaseToNegPow$(-a)_{5}=-a_{5}$

$f(-x)=-x_{5}+3(-x)_{3}−(-x)$

NegBaseToNegPow$(-a)_{3}=-a_{3}$

$f(-x)=-x_{5}+3(-x_{3})−(-x)$

MultPosNeg$a(-b)=-a⋅b$

$f(-x)=-x_{5}−3x_{3}−(-x)$

SubNeg$a−(-b)=a+b$

$f(-x)=-x_{5}−3x_{3}+x$

$f(x)$ | $f(-x)$ | $-f(x)$ |
---|---|---|

$x_{5}+3x_{3}−x$ | $-x_{5}−3x_{3}+x$ | $-x_{5}−3x_{3}+x$ |

We can see above that $-f(x)=f(-x).$ Therefore, $f$ has an **odd** symmetry.