advance algebra PPT Chapter 2 algebra alg

algebra

1. Lesson 1: Definition of Functions Prepared by: Hannaniah S. Jimanga

2. Objectives: At the end of this lesson, the learners are expected to: 1. Apply definitions, properties, axioms, theorems to simplify algebraic expressions, equations and inequalities.

3. Definition. A function from a set to a set is a relation with domain and range that satisfies the following properties: for every element , there is an element such that ; and for all elements and , if and , then . Notations: or   FUNCTION A function is a relation in which for each value of the first component of the ordered pairs, there is exactly one value of the second component.

4. FUNCTION Function as a Rule (expressed by formulas) A function is a rule by which any allowed value of one variable ( the independent variable) determines a unique value of a second variable (the dependent variable). A function is a relation between a dependent and independent variable/s where in for every value of the independent variable , ( x or input), there exists a unique or a single value of the dependent variable , ( y or output).

5. FUNCTION Example. Function as Set of Ordered Pairs The rule for obtaining the unique value of the dependent variable A from the value of the dependent variable r.  

6. FUNCTION Function as Set of Ordered Pairs A function is a set of ordered pair of real numbers such that no two ordered pairs have the same first coordinate and different second coordinate. Any set of ordered pairs is called a relation. A function is a special relation. Remark : NO two or more ordered pairs in must have the same domain .  

7. FUNCTION Example. Function as Set of Ordered Pairs Consider the following relations from to , where and Which of these relations are functions?   Function Not a Function Not a Function Function

8. FUNCTION Function as an Equation The solution set to an equation involving x and y is a set of ordered pairs of the form If there are two ordered pairs with the same first coordinates and different second coordinates then the equation is not a function. Example: 1. 2. 3.   Not a Function Function Not a Function

9. FUNCTION Function shown in Tabular form Tables are used to provide a rule for pairing the value of one variable with the value of another. Each value of the independent variable must correspond to only one value of the dependent variable.

10. FUNCTION Example. Function shown in Tabular form Weight ( lb ) x Cost ($) y 0 to 10 4.60 11 to 30 12.75 31 to 79 32.90 80 to 99 55.82 Weight ( lb ) x Cost ($) y 0 to 15 4.60 10 to 30 12.75 31 to 79 32.90 80 to 99 55.82 x y 1 1 -1 1 -2 2 3 3

11. FUNCTION Function as a graph Every function has a corresponding graph in the xy -plane. NO two or more points must be intersected on the graph of a function upon applying vertical line test. The Vertical-Line Test A graph is the graph of a function if and only if there is no vertical line that crosses the graph more than once

12. (a) (b) (c) (d) NOT A FUNCTION FUNCTION NOT A FUNCTION FUNCTION FUNCTION Consider the following graph, which of them are functions?

13. Domain and range of a Function Domain is the set of all independent values Range is the set of all the dependent values Example: Give the domain and range of the relation and tell whether it defines a function.   Not a Function Domain: {3,4,6} Range: {-1,2,5,8}

14. Domain and range of a Function Domain is the set of all independent values Range is the set of all the dependent values Example: Give the domain and range of the relation and tell whether it defines a function.   Function Domain: { } Range: { }  

15. Domain and range of a Function Domain is the set of all independent values Range is the set of all the dependent values Example: Give the domain and range of the relation and tell whether it defines a function.   Not a Function Domain: { } Range: { }  

16. Kinds of Functions and its graph Linear F unctions A linear function is in the form Where m and b are real numbers with . If then we get → constant function If , then we get → identity function  

17. Kinds of Functions and its graph Linear F unctions Example: Graph the following and state the domain and range.  

18. Kinds of Functions and its graph Absolute value functions The absolute value function is the function defined by Example: Graph the following and state its domain and range   x -2 -1 1 2 3 f(x) 2 1 1 2 3

19. Kinds of Functions and its graph Absolute value functions Example: Graph the following and state its domain and range   x -3 -2 -1 1 2 3 f(x) x -3 -2 -1 1 2 3 f(x)

20. Graphs of functions and relations Quadratic functions A quadratic function is a function of the form W here a, b and c are real numbers, with . Example: Graph the function and state the domain and range. 1.   x -3 -2 -1 1 2 3 f(x)

21. Graphs of functions and relations Quadratic functions Example: Graph the function and state the domain and range.  

22. Graphs of functions and relations Square root functions The square root function is the function defined by Example: Graph and state its domain and range Note: is a real number only when , then   x -3 -2 -1 1 2 3 f(x)

23. Graphs of functions and relations Square root functions Example: Graph the function and state the domain and range.  

24. Graphs of functions and relations Graphing Relations Example: Graph the following and state its domain and range Since the equations expresses x in terms of y, it is easier to choose the white coordinate first.   x y -3 -2 -1 1 2 3