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

Dan Finlay

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

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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:

    • Power functions with odd powers: x to the power of 2 n plus 1 end exponent where n element of straight integer numbers

      • For example: open parentheses negative x close parentheses cubed equals blank minus x cubed

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

      • For example: sin left parenthesis negative x right parenthesis equals negative sin x

    • Linear combinations of odd functions

      • For example: 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:

    • Power functions with even powers: x to the power of 2 n end exponent where n element of straight integer numbers

      • For example: 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

      • For example: cos left parenthesis negative x right parenthesis equals cos x

    • Modulus function: vertical line x vertical line

    • Linear combinations of even functions

      • For example: 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 are the symmetries of graphs of odd & even functions?

  • 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

2-3-3-ib-aa-hl-odd-_-even-functions

Examiner Tips and Tricks

  • Turn your GDC upside down for a quick visual check for an odd function!

    • Ignoring axes, etc, if the graph looks exactly the same both ways, it's odd

Worked Example

a) The graph space y equals f left parenthesis x right parenthesis is shown below. State, with a reason, whether the function space f is odd, even or neither.

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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    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.