Even & Odd Functions (College Board AP® Precalculus): Revision Note

Even & odd functions

What are even functions?

• A function $f\left(x\right)$ is called even if

  • $f(-x)=f\left(x\right)$ for all values of $x$

• A polynomial function $p\left(x\right)=a_n{x}^n$, $n\ge 1$, is even if

  • $\mathit{n}$ is even

    • and $a_n\ne 0$

  • So, e.g., $x^2$, $7x^6$ and $2x^{14}$ are all even functions

• Linear combinations of even functions are also even functions

  • E.g. $x^2+7x^6-2x^{14}$ is an even function

• Not all even functions are polynomial functions

  • E.g. $\cos (-x)=\cos x$

  • So $\cos x$ and $\sec x$ are both even functions

What are odd functions?

• A function $f\left(x\right)$ is called odd if

  • $f(-x)=-f\left(x\right)$ for all values of $x$

• A polynomial function $p\left(x\right)=a_n{x}^n$, $n\ge 1$, is odd if

  • $\mathit{n}$ is odd

    • and $a_n\ne 0$

  • So, e.g., $x$, $4x^5$ and $3x^{17}$ are all odd functions

• Linear combinations of odd functions are also odd functions

  • E.g. $x-4x^5+3x^{17}$ is an odd function

• Not all odd functions are polynomial functions

  • E.g. $\sin (-x)=-\sin x$

  • So $\sin x$ and $\csc x$ are both odd functions

  • $\tan x$ and $\cot x$ are also both odd functions

What do the graphs of even or odd functions look like?

• An even function is graphically symmetric over the line $\mathit{x}\mathbf{=}\mathbf{0}$

  • This means that its graph is unchanged by a reflection in the ­y-axis

  • If a point $\left(x,y\right)$ is on the graph, then the point $\left(-x,y\right)$ is also on the graph

• An odd function is graphically symmetric about the point $(0,0)$

  • This means that for every point on its graph, there is a corresponding point at an equal distance but in the opposite direction through the origin

  • The graph is unchanged by a 180° rotation about the origin

  • If a point $\left(x,y\right)$ is on the graph, then the point $\left(-x,-y\right)$ is also on the graph

How do local maximum and minimum points appear on graphs of even and odd functions?

• The symmetry of even and odd functions means there are correspondences between local maximum and minimum points on either side of the $y$-axis

• For an even function $f$

  • If $\left(x,f\left(x\right)\right)$ is a local maximum then $\left(-x,f\left(-x\right)\right)$ is also a local maximum

  • If $\left(x,f\left(x\right)\right)$ is a local minimum then $\left(-x,f\left(-x\right)\right)$ is also a local minimum

    • In both cases, $f\left(-x\right)=f\left(x\right)$

• For an odd function $g$

  • If $\left(x,g\left(x\right)\right)$ is a local maximum then $\left(-x,g\left(-x\right)\right)$ is a local minimum

  • If $\left(x,g\left(x\right)\right)$ is a local minimum then $\left(-x,g\left(-x\right)\right)$ is a local maximum

    • In both cases, $g\left(-x\right)=-g\left(x\right)$