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Even & Odd Functions (College Board AP® Precalculus): Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Even & odd functions

What are even functions?

  • A function space f left parenthesis x right parenthesis is called even if

    • space f left parenthesis negative x right parenthesis equals f left parenthesis x right parenthesis for all values of x

  • A polynomial function space p open parentheses x close parentheses equals a subscript n x to the power of n, space n greater or equal than 1, is even if

    • bold italic n is even

      • and a subscript n not equal to 0

    • So, e.g., x squared, 7 x to the power of 6 and 2 x to the power of 14 are all even functions

  • Linear combinations of even functions are also even functions

    • E.g. space x squared plus 7 x to the power of 6 minus 2 x to the power of 14 is an even function

  • Not all even functions are polynomial functions

    • E.g. cos left parenthesis negative x right parenthesis equals cos x

    • So cos x and sec x are both even functions

What are odd functions?

  • A function space f left parenthesis x right parenthesis is called odd if

    • space f left parenthesis negative x right parenthesis equals negative f left parenthesis x right parenthesis for all values of x

  • A polynomial function space p open parentheses x close parentheses equals a subscript n x to the power of n, space n greater or equal than 1, is odd if

    • bold italic n is odd

      • and a subscript n not equal to 0

    • So, e.g., x, 4 x to the power of 5 and 3 x to the power of 17 are all odd functions

  • Linear combinations of odd functions are also odd functions

    • E.g. space x minus 4 x to the power of 5 plus 3 x to the power of 17 is an odd function

  • Not all odd functions are polynomial functions

    • E.g. sin left parenthesis negative x right parenthesis equals negative 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 bold italic x bold equals bold 0

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

    • If a point open parentheses x comma space y close parentheses is on the graph, then the point open parentheses negative x comma space y close parentheses is also on the graph

  • An odd function is graphically symmetric about the point stretchy left parenthesis 0 comma space 0 stretchy right parenthesis

    • 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 open parentheses x comma space y close parentheses is on the graph, then the point open parentheses negative x comma space minus y close parentheses is also on the graph

Graphs illustrating odd and even functions. Odd functions are unchanged by 180° rotation, and even functions are unchanged by reflection in the y-axis.

Examiner Tips and Tricks

Rotating your graphing calculator by 180° can help to check if a graph is odd!

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 space y-axis

  • For an even function space f

    • If open parentheses x comma space f open parentheses x close parentheses close parentheses is a local maximum then open parentheses negative x comma space f open parentheses negative x close parentheses close parentheses is also a local maximum

    • If open parentheses x comma space f open parentheses x close parentheses close parentheses is a local minimum then open parentheses negative x comma space f open parentheses negative x close parentheses close parentheses is also a local minimum

      • In both cases, space f open parentheses negative x close parentheses equals f open parentheses x close parentheses

  • For an odd function space g

    • If open parentheses x comma space g open parentheses x close parentheses close parentheses is a local maximum then open parentheses negative x comma space g open parentheses negative x close parentheses close parentheses is a local minimum

    • If open parentheses x comma space g open parentheses x close parentheses close parentheses is a local minimum then open parentheses negative x comma space g open parentheses negative x close parentheses close parentheses is a local maximum

      • In both cases, space g open parentheses negative x close parentheses equals negative g open parentheses x close parentheses

Worked Example

The polynomial function space p is an odd function. If space p open parentheses 7 close parentheses equals 5 is a relative minimum of space p, which of the following statements about space p open parentheses negative 7 close parentheses must be true?

(A) space p open parentheses negative 7 close parentheses equals 5 is a relative minimum

(B) space p open parentheses negative 7 close parentheses equals negative 5 is a relative minimum

(C) space p open parentheses negative 7 close parentheses equals 5 is a relative maximum

(D) space p open parentheses negative 7 close parentheses equals negative 5 is a relative maximum

Answer:

An odd function is graphically symmetric about the point open parentheses 0 comma space 0 close parentheses

  • This means that space p open parentheses negative x close parentheses equals negative p open parentheses x close parentheses

    • i.e. space p open parentheses negative 7 close parentheses equals negative p open parentheses 7 close parentheses equals negative 5

  • And also a minimum at space p open parentheses x close parentheses corresponds to a maximum at space p open parentheses negative x close parentheses, and vice versa

    • i.e. space p open parentheses 7 close parentheses is a relative minimum, so space p open parentheses negative 7 close parentheses is a relative maximum

(D) space p open parentheses negative 7 close parentheses equals negative 5 is a relative maximum

Examiner Tips and Tricks

In the Worked Example, don't be fooled by the fact that space p open parentheses negative 7 close parentheses less than p open parentheses 7 close parentheses. That doesn't stop space p open parentheses 7 close parentheses equals 5 from being a relative (i.e. local) minimum and space p open parentheses negative 7 close parentheses equals negative 5 from being a relative (i.e. local) maximum.

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.