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Functions

If a unique real number y is associated to each value of a real variable x by means of a rule f, then we say the variable y is a real-valued function of the real variable x. This is denoted by y = f (x). The variable x is called the independent variable and y is called the dependent variable. It means that if we assign any value for x, we will get a unique value of y governed by the function f (x). Let us understand this by an example.

 

Example
If y = f(x= 2x2 + 3x – 1, find the values of y corresponding to x = 1, 0 and -1.
Solution
We can find the required values of y by substituting given values of x in the function.
Value of y at x = 1 will be f(1) = 2(1)2 + 3(1) - 1 = 2 + 3 – 1 = 4
Value of y at x = 0 will be f(0) = 2(0)2 + 3(0) - 1 = 0 + 0 – 1 = -1
Value of y at x = -1 will be f(-1) = 2(-1)2 + 3(-1) - 1 = 2(1) - 3 - 1 = -2
 
Example
If f(x= |x+ |x + 4| then redefine the function in simple terms and find the values of: f(2), f(-2), and f(-5).
Solution
|x| represents the ‘modulus function’ or the ‘absolute value function.’ It gives the absolute value (positive value) of the quantity inside the function denoted by two parallel bars | |. So, if we have |2|, it will give us 2, while |-2| will also give us 2. Similarly,
 
|x| = x when x > 0     and   |x| = -x when x < 0
 
|x + 4| = x + 4 when x + 4 > 0 or x> -4   and  |x + 4| = -(x + 4) when x + 4 < 0 or x < -4
 
Combining the above two conditions, we can redefine f(x) as,
For x > 0, f(x) = x + x + 4 ⇒ f(x) = 2x + 4
 
For -4 < x < 0, f(x) = -x + x + 4 ⇒ f(x) = 4
 
For x < -4, f(x) = -x + -(x + 4) ⇒ f(x) = -2x – 4
 
Now, since 2 > 0, f(2) = 2(2) + 4 = 8
 
-2 lies between -4 and 0 hence, f(-2) = 4
 
-5 < -4 so, f(-5) = -2(-5) - 4 = 6. 

Types of Function

Even and odd functions
  1. Even Functions: A function f (x) is said to be an even function if f (-x= f (x).
     

    Examples: f(x= x2 + 2x4

     

    f(–x= (-x )2 + 2 (-x )4 = x2 + 2x4 = f(x)

     

    Hence, f(x= x2 + 2x4 is an even function.

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

    Examples: f (x= 5x + 6x3

     

    f (-x= 5(-x ) + 6(-x)3 = -5x - 6x3 = -(5x + 6x3)

     

    Hence, 5x + 6x3 is an odd function.

Periodic functions

A function f (x) 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 then it is periodic function. Symbolically, if f (x + p= f (x) for all x, then p is the period of f (x).





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