1. Intro to Algebra 2 Summary

2. Intro & Summary This chapter introduces relations and functions. Functions will be the focus of most of the rest of algebra, as well as pre-calculus and calculus. This chapter is an important stepping stone to the rest of algebra. The first section deals with the difference between relations and functions. It explains how to represent relations and functions using both mapping diagrams and graphs. It also explains how to determine whether or not a relation is a function, given a representation of that relation. The second section deals with domains of functions and relations; that is, the set of values which are inputs for a relation or function. It explains how to determine the domain of a relation. It also deals with two types of restricted domain--restrictions of an infinite set of numbers, and restrictions of a few points. Some functions cannot take certain values as inputs, and this section details how to find those values. The final section deals with ranges of functions and relations. While domain is the set of inputs of a function, range is the set of outputs. Both are important to mention when describing a graph or function. Much of Algebra II and Calculus is concerned with the study of the properties of functions. Here, the student will graph functions, find their maximum and minimum values, and determine their attributes solely from their equations. Functions will be used to solve many different types of problems. However, one must first learn the basics--how to recognize a function, and how to determine its domain and range.

3. Terms Domain - The set of all inputs of a relation or function. Function - A relation in which each input has only one output. Often denoted f (x). Horizontal Line Test - If every horizontal line you can draw passes through only 1 point, x is a function of y. If you can draw a horizontal line that passes through 2 points, x is not a function of y. Range - The set of all outputs of a relation or function. Relation - A set of inputs and outputs, often written as ordered pairs (input, output). Restricted Domain - The domain of a function that is not defined for all real numbers. Vertical Line Test - If every vertical line you can draw goes through only 1 point, y is a function of x. If you can draw a vertical line that goes through 2 points, y is not a function of x.

4. Relation & Functions Relation: A relation is simply a set of ordered pairs. Function: A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.

5. Relations A relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping diagram or a graph. For example, the relation can be represented as: Mapping Diagram of Relation Lines connect the inputs with their outputs. The relation can also be represented as: Graph of Relation

6. Functions A function is a relation in which each input has only one output. Examples: y is a function of x, x is a function of y. y is not a function of x ( x = 3 has multiple outputs), x is a function of y. y is a function of x, x is not a function of y ( y = 9 has multiple outputs). y is not a function of x ( x = 1 has multiple outputs), x is not a function of y ( y = 2 has multiple outputs).

7. The Line Test for Mapping Diagrams To check if a relation is a function, given a mapping diagram of the relation, use the following criterion: If each input has only one line connected to it, then the outputs are a function of the inputs. Example: In the following mapping diagram, y is a function of x, but x is not a function of y :

8. The Vertical and Horizontal Line Tests for Graphs To determine whether y is a function of x, given a graph of a relation, use the following criterion: if every vertical line you can draw goes through only 1 point, y is a function of x. If you can draw a vertical line that goes through 2 points, y is not a function of x. This is called the vertical line test. Example 1: In the following graph, y is a function of x :

9. Example 2: In the following graph, y is not a function of x. Fails Vertical Line Test

10. Domain The domain of a relation (or of a function) is the set of all inputs of that relation. For example, the domain of the relation (0, 1),(1, 2),(1, 3),(4, 6) is x = 0, 1, 4.

11. Restricted Domain Is the graph a function? What is the domain of the following graph? Graph

12. Domain The domain of the following mapping diagram is -2, 3, 4, 10 : Mapping Diagram

13. Domain Most of the functions we have studied in Algebra I are defined for all real numbers. But some functions have restrictions on their domain. In general, there are two types of restrictions on domain: 1.restrictions of an infinite set of numbers, and 2.restrictions of a few points. Square root signs restrict an infinite set of numbers, because an infinite set of numbers make the value under the sign negative. To find the domain of a function with a square root sign, set the expression under the sign greater than or equal to zero, and solve for x.

14. Example Find the domain of each function below:

15. Problems Problem : What is the domain of the relation (5, 4),(2, 3),(2,-4),(4,-1),(0, 2)? Problem : What is the domain of the relation in the following mapping diagram?

16. Problems Problem : What is the domain of the relation in the following graph?

17. Problems Problem : What is the domain of the relation in the following graph?

18. Problem Problem : What is the domain of each of the following functions? a) f (x) = 4x - 16 b) f (x) = c) f (x) = d) f (x) = (x + 4) 2 e) f (x) =

19. Problem : What is the domain of each of the following functions? a) f (x) = b) f (x) = c) f (x) = d) f (x) = e) f (x) =

20. Range The range of a relation (or function) is the set of all outputs of that relation. The range of the following mapping diagram is -1, 0, 2, 4, & 6.

21. What is the range of the following function?

22. Problem What is the range of the relation (3, 6),(-2,- 4),(6, 1),(2, 4),(3,-4) ?

23. Problem What is the range of the relation in the following mapping diagram?

24. Problem What is the range of the relation in the following graph?

25. Problem What is the range of the relation in the following graph?

26. Problem What is the range of the function in the following graph?