Mathematical Functions Definition

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Definition: Let X and Y be the two sets. A function f from set X to set Y is a rule of associating elements of set X to elements of set Y such that every element in xЄ X is uniquely associated with some element in yЄ Y.

Notation: f: X→Y

Image:


Definition: The element y is called as the image of x under the function f

Notation: y = f(x)

Where x is called as the pre – image of y.

(Show by diagram)

Domain and co domain:

Definition: If f: X→Y , then the set X is domain of f and set Y is called as co domain of f.

Range:

Definition: The range of a function f: X Y is the set of all values taken by function f i.e. Iit is the set of all images of the elements of X .

Notation: Range = {f(x), xЄX }
                             Clearly f(x)⊂Y .


Types of function:

Even function: A function f(x) is said to be an even function if f (-x) = f(x) for all x.

Example: y = x2 is an even function as (-x)2 = x2.

Odd function: A function f(x) is said to be an odd function if f (-x) = -f(x) for all x.

Example: y = x3 is an odd function of x as (-x)3 = -x3 = -f(x).

Periodic function : A function in which the range of the independent variable can be separated into equal sub intervals such that the graph of the function is the same in each part is the period i.e. f(x+a) = f(x) for all x then a is the period of f.

a. 11 -2h
b. 11+2h
c. 2h -11
d. none of these

 Hence the correct option is b).

Hence the correct option is d).

Example 3: If f(x) = x2 - |x| then f(x) is

a. odd function
b. periodic function
c. even function
d. Composite function.

Solution: Given: f(x) = x2 - |x| then f( -x) = (-x2) - |-x|
                                = x2 - |x| = f(x)
                                   f(x) is an even function.


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