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.
We may assume that the function f(x) is an even function. The complete graph for this function will look as follows:
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:
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) + k causes a translation of k units parallel to the y-axis.
A function:
g(kx)
will be an even function. The algebraic transformation g(kx) causes a stretch parallel to the x-axis by a scale factor of (1/a).
A function:
kg(x)
will be an even function. The algebraic transformation kg(x) causes a stretch parallel to the y-axis by a scale factor of a.
As stated above, this may be generalised and conclusions about transformations of even functions may be drawn. These are:
-
an even function is symmetrical about the y-axis;
- an even function cannot be one-to-one function;
-
the function remains even, although the scale factor a of an even function is changed;
-
when an even function is translated parallel to the x-axis, after translation the function is no longer even;
-
when an even function is translated parallel to the y-axis, after translation the function remains even.
If we consider an unknown function h(x) which is odd, certain other functions will be also odd for k ≠ 0, and thus certain conclusions can be drawn from it.
A function:
h(x + k)
will not be an odd function. The algebraic transformation h(x + k) causes a translation of -k units left parallel to the x-axis.
A function:
h(x) + k
will not be an odd function. The algebraic transformation h(x) + k causes a translation of k units parallel to the y-axis.
A function:
h(kx)
will be an odd function. The algebraic transformation h(kx) causes a stretch parallel to the x-axis by a scale factor of (1/a).
A function:
kh(x)
will be an odd function. The algebraic transformation kg(x) causes a stretch parallel to the y-axis by a scale factor of a.
As stated above, this may be generalised and conclusions about transformations of odd functions may be drawn. These are:
- an odd function is symmetrical about the origin;
-
the function remains odd, although the scale factor a of an odd function is changed;
-
when an odd function is translated parallel to the x-axis, after translation the function is no longer odd;
-
when an odd function is translated parallel to the y-axis, after translation the function is no longer odd.
If we have an inverse of any even function, the graph obtained will not be a function. There will be two possible arguments for each value.
The inverse is illustrated by an example below:
If we have an inverse of any odd function, the graph obtained will also be a function. In some cases the reflection will cover the function.
The inverse is illustrated by an example below:
It is possible for an even function to have a domain , but restricted range. An example of such a function may be already sketched graph of f (x) = x2.
The domain of a function f (x) = x2 is , while the range is 0 ≤ f(x) ≤ ∞.
It is impossible for any negative number to be included. The explanation can be found on page 2.
It is possible for an odd function to have a domain , but restricted range. An example of such a function may be already sketched graph of
The domain of a function is , while the range is 0 ≤ f(x) ≤ ∞.
It is an asymptote, therefore the curves will never reach zero. In this case zero is the only number that must be excluded from the range.
To sum up, I am to gather all information and conclusions I managed to draw.
Sources:
-
Robert Smedley, Garry Wiseman. Mathematics Standard Level. United Kingdom, Oxford Univeristy Press, 2004.
- http://www.mathematicshelpcentral.com/lecture_notes/precalculus_algebra_folder/odd_and_even_functions.htm
- http://en.wikipedia.org/wiki/Even_and_odd_functions
- http://www.purplemath.com/modules/fcnnot3.htm
