An even function exists when for all values of x.
The graph of an even function must be symmetrical about the y-axis.
Examples of even functions have been listed below:
An odd function exists when for all values of x.
Graphs of odd functions should have 180 degrees rotational symmetry about the origin (0,0).
Examples of odd functions are listed below:
Most functions are neither even or odd.
An example of a function that is neither even or odd would be:
Commutative means that the order of elements contained within an expression do not alter a certain mathematical result.
Knowing this, you can say that:
Associative means that you can alter the grouping of certain elements in an expression without changing its result.
Knowing this you could say that:
Commutative and Associative manipulation should not be used when subtracting and dividing.
Notice that what we’re ultimately doing in both cases is multiplying the surd within a fraction by 1. When the value of the numerator is exactly the same as the value of the denominator in a fraction, what you have is 1.
You should know that 1/1, 2/2, 3/3, (a+b)/(a+b) are all equal to 1.
In order to multiply surds, you should first know these rules:
You should also know that:
So, knowing these rules, what would you get if you multiplied: ?
Now how about ?
What about ?
Firstly, you need to know what the product rule is:
Now integrate each term with respect to x: