Mathematic Formulas and important concept


Chapter 1: Function

Definition: function is relation of object and image. In other word, function is a correspondence between two sets A and B that assigns to each element of set A to element of B:
Denoted by: f : A -->B or f(a)=b
What is object and image?

a. Relation
One-to-one: Each element of set A for each element of set B

One-to-many: More image(element of set B) of one object(element of set A)

Many-to-many:


**A is called domain of the function f and b is called the codomain. The image of A is called range of f




b. Composite function

Definition: Combination of two functions

**if f:A-->B and g:B-->C, thus composite gf is function of gf: A--> C
**gf(a)=g(f(x))

c. Inverse function:

Definition: if domain to codomain is called function, then codomain refer to domain is called inverse function.


**inverse function of f is f¯¹
**if f(a)= b, then f¯¹(b) = a






d. Example question:

1. Given f(x) = 2x and g(x) = 3x+2, find gf(x) and fg(x).

gf(x) = g(f(x))
= g(2x)
= 3(2x) +2
= 6x + 2
fg(x) = f(g(x))
= f(3x+2)
= 2(3x + 2)
= 6x + 4
2. Given f(x)= 2x + 1, find f¯¹(x).

Let f(y) = x----------1
F(y) = 2y + 1--------2
2 into 1:
2y + 1 = x
2y = x - 1
y= (x-1)/2
thus, f¯¹(x) =(x-1)/2

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