FUNCTION

FUNCTION

Domain:

It is a set of values for which the function has been defined.

(OR)

For a function y = f(x), the set of values of x are called domain, and set of y values are called Range.

f(x) 2x+3

Here x is a parameter

Example: with f(x) = x²

an input of 4

becomes an output of 16.

In fact we can write f(4) = 16.

Co-domin:

It is set of values which can be written by f.

Special Definitions:

1. One-One function: Every element of the range of the function corresponds to exactly one element of the domain.

2. Onto function: Co-domain equals to range. It means that every element in the co-domain is part of the mapping.

3. Bijection: If a function is both one-one and onto, it is called a bijection.

INVERSE of a function:

f(x)=3x+4

Solution : f(x)=3x+4=y

   3x=y-4

  x=(y-4)/3

=>  f^-1(y)=(y-4)/3 => f^-1(x)=(x-4)/3

Even and Odd function:

Even function: If f(x) = f(-x) then we call that function as Even function. Even functions are symmetric around Y axis

Example : f(x)=2x² +5

=> f(-x)=2(-x²)+5=2x²+5=f(x)

Odd function: If f(-x) = -f(x) then we call that function as Odd function. In the graph of odd function, the first and third quadrants will be reflections of each other and so will the second and fourth quadrant.

Example :  g(x)=3x³+5x

=> g(-x)= 3(-x)³+5(-x)= -3x³-5x= -(3x³+5x)= -g(x)

There is neither even nor odd function

h(x)= 3x²+4x=5

=> h(-x)= 3x²-4x+5

Composite functions:

If we have f(x)= 2x+1 , g(x)= x²+5.

Calculate g(f(2))...

=> put x=2 in f(x) = 2(2)+1= 5

Then we have g(5). Now put x=5 in g(x)

=> g(5)= 5²+5= 30

Piece-wise defined function:

 The domain of y= 9-√(9-x²) is:

 A. x< -3

 B. -3<x<3

 C. x ∈ [-3,3]

 D. none of the above

Solution: f(x)= 9-√(9-x²)

=> 9-x² ≥ 0

=> x²-9 = 0

=> (x-3)(x+3)

=> x ∈ [-3,3]