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Odd and Even Functions Portfolio

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Introduction

Maths SL

Type I Portfolio

Topic:         Odd & Even Functions.

Aim:                 To investigate the symmetry of odd and even functions.

Author:         Aleksandra Rodzik

Certain functions are classified as “odd” or “even” functions on the basis of their symmetry in the Cartesian coordinate plane and corresponding algebraic properties.  The purpose of this investigation is to explore the characteristics of these functions and transformations of these functions.

Algebraically, even and odd functions are defined as follows:

  • An even function is defined as a function image00.png for which image01.png.
  •  An odd function is defined as a function image00.png for which image13.png.

Each elementary function can be classified as odd, even or neither. This is illustrated by some examples below.

even

odd

neither

image14.png

image15.png

f (x) = x2

f (x) = x3

f (x) = x

image11.png

image16.png

f (x) = 1

image17.png

f (x) = 2x

To verify whether a given function is even, odd or neither, you have to plug  –x for x and simplify. If the result obtained is exactly the same as the formula given, then the function is even (f(–x) = f(x)). If the result is the opposite of the formula, then the function is odd (f(–x) = –f(x)). Finally, when the result is utterly different from the formula given, the function is neither even nor odd.

The examples will be now followed by calculations.

f(x) = |x|

f(-x) = |x|

f(x) = f(–x)

even function

=>

As the absolute value cannot be negative

f(x) = √(1-x2)

f(-x) = √[1-(-x2)]

f(x) = f(–x)

even function

=>

As raising –x to the power of 2 gives x2

f(x) = x2

f(-x) = (-x)2

f(-x) = x2

even function

=>

As above

f(x) = x3

f(-x) = (-x)3

f(-x) = -x3

odd function

=>

As raising –x to the power of 3 gives -x3

f(x) = x

f(-x) = -x

odd function

f(x) = (1/x)

f(-x) = [1/(-x)]

f(-x) = -(1/x)

odd function

=>

As dividing by –x changes the sign

...read more.

Middle

=>

As it is impossible to square a negative number

f(x) = 1

f(-x) = 1

neither

=>

As there is no x to substitute

image17.png

neither

image02.png

f(x) = 2x

f(-x) = 2-x

f(-x) = (1/2)x

neither

As shown above, even functions are symmetrical about the y-axis, while odd functions are symmetrical about the origin.

I would like to consider a certain graph of a function f(x) which is shown below.

image03.png

We may assume that the function f(x) is an even function. The complete graph for this function will look as follows:

image04.png

The range of the completed function will be therefore the same for both graphs:

0 ≤ f(x) ≤ 7

We may also assume that the function f(x) is an odd function. The complete graph for this function will look as follows:

image05.png

The range of the completed function will be therefore different for both graphs. For f(x) as an odd function it will reach:

        -7 ≤ f(x) ≤ 7

If we consider an unknown function g(x) which is even, certain other functions will be also even for k ≠ 0, and thus certain conclusions can be drawn from it.

A function:

g(x + k)

will not be an even function. The algebraic transformation g(x + k) causes a  translation of -k units left parallel to the x-axis.

A function:

g(x) + k

will be an even function. The algebraic transformation g(x) +

...read more.

Conclusion

To sum up, I am to gather all information and conclusions I managed to draw.

even functions

odd functions

  1. defined as a function image00.png for which image01.png.
  2. symmetrical about the y-axis;
  3. cannot be one-to-one function;
  4. the function remains even, although the scale factor a of an even function is changed;
  5. when an even function is translated parallel to the x-axis, after translation the function is no longer even;
  6. when an even function is translated parallel to the y-axis, after translation the function remains even;
  7. the inverse of any even function will not be a function;
  8. It is possible for an even function to have a domain image10.png, but restricted range.
  1. image00.png for which image13.png;
  2. symmetrical about the origin;
  3. the function remains odd, although the scale factor a of an odd function is changed;
  4. when an odd function is translated parallel to the x-axis, after translation the function is no longer odd;
  5. when an odd function is translated parallel to the y-axis, after translation the function is no longer odd;
  6. an inverse of any odd function will also be a function;
  7. It is possible for an odd function to have a domain image10.png, but restricted range;

Sources:

  1. Robert Smedley, Garry Wiseman. Mathematics Standard Level. United Kingdom, Oxford Univeristy Press, 2004.
  2. http://www.mathematicshelpcentral.com/lecture_notes/precalculus_algebra_folder/odd_and_even_functions.htm
  3. http://en.wikipedia.org/wiki/Even_and_odd_functions
  4. http://www.purplemath.com/modules/fcnnot3.htm

...read more.

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