functions.pdf - Functions Function: A function is a...

Functions
Function: A function is a relationship between two variables (one is called independent
variable, another is called dependent variable) such that to each value of the independent
variable there corresponds exactly one value of the dependent variable .
• Domain of the function y = f ( x ): D = The set of all values of the independent variable
x for which the function is defined.
• Range of the function: R = The set of all values taking on by the dependent variable
y .
Four ways to represent functions:
• Verbally: by a description in words;
• numerically: by a table of values;
• visually: by a graph { ( x, f ( x )) | x ∈ D } ;
• algebraically: by an explicit formula.
Example 1 f ( x ) =
x
2
x 2 - 3 x +2
is a function, D = { x : x 6 = 1 , 2 } .
Example 2 f ( x ) =
√
4 - x
2
is a function, D = { x : - 2 ≤ x ≤ 2 } .
Example 3 f ( x ) = ± x 2 is not a function.
Example 4 Piecewise defined functions: f ( x ) =
(
2 x, x ≤ 0 ;
3 x, x > 0 .
Example 5 Evaluate a function: Let f ( x ) = x
2
- 2 x + 1 . Then f (5) = 5
2
- 2(5) + 1 = 16 .
f ( x + 3) = ( x + 3)
2
- 2( x + 3) + 1 = x
2
+ 4 x + 4 .