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Odd & Even Functions (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Mark Curtis

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

Odd & 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

  • Examples of odd functions include

    • Odd powers,x to the power of 2 n plus 1 end exponent where n element of straight integer numbers

      • e.g. open parentheses negative x close parentheses cubed equals blank minus x cubed

    • Some trig functions: sin x, cosec x, tan x, cot x

      • e.g. sin left parenthesis negative x right parenthesis equals negative sin x

    • Linear combinations of odd functions

      • e.g. space f open parentheses x close parentheses equals 3 x to the power of 5 minus 4 sin invisible function application x plus 6 over x

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

  • Examples of even functions include

    • Even powers x to the power of 2 n end exponent where n element of straight integer numbers

      • e.g. open parentheses negative x close parentheses to the power of 4 equals blank x to the power of 4

    • Some trig functions: cos x, sec x

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

    • Modulus function vertical line x vertical line

    • Linear combinations of even functions

      • e.g. space f open parentheses x close parentheses equals 7 x to the power of 6 plus 3 open vertical bar x close vertical bar minus 8 cos invisible function application x

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

  • The graph of an odd function has rotational symmetry

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

  • The graph of an even function has reflective symmetry

    • The graph is unchanged by a reflection in the ­y-axis

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

You can rotate your GDC by 180° to check if a graph is odd!

Worked Example

(a) The graph y equals f left parenthesis x right parenthesis is shown below. State, with a reason, whether the function space f is odd or even.

2-3-3-ib-aa--ai-we-image-a
2-3-3-ib-aa-hl-odd-even-functions-a-we-solution

(b) Use algebra to show that g open parentheses x close parentheses equals x cubed sin invisible function application open parentheses x close parentheses plus 5 cos invisible function application open parentheses x to the power of 5 close parentheses is an even function.

2-3-3-ib-aa-hl-odd-even-functions-b-we-solution

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    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.