DP IB Maths: AA HL

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IBMaths: AA HLDPRevision Notes2. Functions2.3 Functions Toolkit2.3.3 Symmetry of Functions

2.3.3 Symmetry of Functions

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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

Exam Tip

  • 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

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

Exam Tip

  • 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

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

Exam Tip

  • 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

    Author: Dan

    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.