Mathematical Functions Definition

The below post by the team of homeworkhelpexperts.com would be useful for our readers to understand about the basics of Mathematical Functions.

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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Differentiation of a Function

Derivative or differential coefficient of the function is an important branch of mathematics. Let y = f(x) be the real valued function. It means the value of y depends on the value of x and it changes with a change in the value of x.  So, x is called the independent variable and y is the dependent variable.  In simple terms, derivative of a function denotes how a quantity is changing with respect another quantity. 

Let ∆ x (or h) be the small increment in x and the corresponding change in y or f(x) be

∆y.  Then the value of  x  changes  from x  to x + ∆ x ( or h ) and the value of f(x) or y  changes from   f( x )  to f ( x + ∆ x).

So, change in the value of f(x) or ∆ y = f (x + ∆ x) – f(x).

Definition of derivatives or differentiation or differential coefficient of a function f (x):

 

Differentiation from first principles or ab – initio method or delta method:

The process of finding the derivative of a function   by using the definition of derivatives is called the differentiation from first principles. Using the first principle of differentiation, mathematicians have derived some standard formulas based on them; it becomes very easy to calculate differentiation of most of the functions.

Standard formulas

 

 

By using the above standard formulas, derivative of a function can be calculated as given below:

Some of the examples of solved problems are as below:

 Example 1:            x^y  =y^x

Solution:         Taking log on both the sides

           y log x = x log y

Differentiating both sides with respect to x:

Above is the final answer of the derivative of y with respect to x.

We hope that this post would help our students understand about what is derivative of a function and how can one calculate the derivative or differentiation of a function. If you need math homework help or math assignment help, please email us to info@homeworkhelpexperts.com