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Symmetry of Functions (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

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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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Periodic Functions

What are periodic functions?

  • A function space f left parenthesis x right parenthesis is called periodic, with period k, if

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

  • Examples of periodic functions include:

    • sin x & cos x: The period is 2π or 360° 

    • tan x: The period is π or 180°

    • Linear combinations of periodic functions with the same period

      • For example: f open parentheses x close parentheses equals 2 sin invisible function application open parentheses 3 x close parentheses minus 5 cos invisible function application open parentheses 3 x plus 2 close parentheses

What are the symmetries of graphs of periodic functions?

  • The graph of a periodic function has translational symmetry

    • The graph is unchanged by translations that are integer multiples of stretchy left parenthesis table row k row 0 end table stretchy right parenthesis

    • The means that the graph appears to repeat the same section (cycle) infinitely

2-3-3-ib-aa-hl-periodic-functions

Examiner Tips and Tricks

  • There may be several intersections between the graph of a periodic function and another function

    • i.e.  Equations may have several solutions so only answers within a certain range of x-values may be required

      • e.g. Solve  tan space x equals square root of 3  for  0 degree space less or equal than space x less or equal than space 360 degree

      • x equals 60 degree comma space 240 degree

    • Alternatively you may have to write all solutions in a general form

      • e.g.  x equals 60 left parenthesis 3 k plus 1 right parenthesis degree comma space space space k equals 0 comma space plus-or-minus 1 comma space plus-or-minus 2 comma space...

Worked Example

The graph space y equals f left parenthesis x right parenthesis is shown below. Given that space f is periodic, write down the period.

2-3-3-ib-aa--ai-we-image-c
2-3-3-ib-aa-hl-periodic-functions-we-solution

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Self-Inverse Functions

What are self-inverse functions?

  • A function space f left parenthesis x right parenthesis is called self-inverse if

    • left parenthesis f ring operator f right parenthesis left parenthesis x right parenthesis equals x for all values of x

    • space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals f left parenthesis x right parenthesis

  • Examples of self-inverse functions include:

    • Identity function: space f left parenthesis x right parenthesis equals x

    • Reciprocal function: space f left parenthesis x right parenthesis equals 1 over x

    • Linear functions with a gradient of -1: space f left parenthesis x right parenthesis equals negative x plus c

What are the symmetries of graphs of self-inverse functions?

  • The graph of a self-inverse function has reflective symmetry

    • The graph is unchanged by a reflection in the line y = x

2-3-3-ib-aa-hl-self-inverse-functions

Examiner Tips and Tricks

  • If your expression for  f to the power of negative 1 end exponent left parenthesis x right parenthesis  is not the same as the expression for  f left parenthesis x right parenthesis  you can check their equivalence by plotting both on your GDC

    • If equivalent the graphs will sit on top of one another and appear as one 

    • This will indicate if you have made an error in your algebra, before trying to simplify/rewrite to make the two expressions identical

  • It is sometimes easier to consider self inverse functions geometrically rather than algebraically

Worked Example

Use algebra to show the function defined by space f open parentheses x close parentheses equals fraction numerator 7 x minus 5 over denominator x minus 7 end fraction comma blank x not equal to 7 is self-inverse.

2-3-3-ib-aa-hl-self-inverse-functions-we-solution

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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.