1. Functions (Domain, Range, Composition)

2. Symbols for Number Set Counting numbers ( maybe 0, 1, 2, 3, 4, and so on) Natural Numbers: Positive and negative counting numbers (-2, -1, 0, 1, 2, and so on) Integers: a number that can be expressed as an integer fraction (-3/2, -1/3, 0, 1, 55/7, 22, and so on) Rational Numbers: a number that can NOT be expressed as an integer fraction (π, √2, and so on) Irrational Numbers: NONE

3. Symbols for Number Set The set of all rational and irrational numbers Real Numbers: Natural Numbers Integers Rational Numbers Irrational Numbers Real Number Venn Diagram:

4. Set Notation Not Included The interval does NOT include the endpoint(s) Interval NotationInequality NotationGraph Parentheses ( ) < Less than > Greater than Open Dot Included The interval does include the endpoint(s) Interval NotationInequality NotationGraph Square Bracket [ ] ≤ Less than ≥ Greater than Closed Dot

5. Graphically and algebraically represent the following: All real numbers greater than 11 Graph: Inequality: Interval: Example 1 10 11 12 Infinity never ends. Thus we always use parentheses to indicate there is no endpoint.

6. Describe, graphically, and algebraically represent the following: Description: Graph: Interval: Example 2 1 3 5 All real numbers greater than or equal to 1 and less than 5

7. Describe and algebraically represent the following: Describe: Inequality: Symbolic: Example 3 -2 1 4 All real numbers less than -2 or greater than 4 The union or combination of the two sets.

8. Functions Algebraic Function Can be written as finite sums, differences, multiples, quotients, and radicals involving x n. Examples: Transcendental Function A function that is not Algebraic. Examples: A relation such that there is no more than one output for each input ABCABC WZWZ

9. Domain and Range Domain All possible input values (usually x), which allows the function to work. Range All possible output values (usually y), which result from using the function. The domain and range help determine the window of a graph. x y f

10. Example 1 Domain: Range: Domain: Range: Describe the domain and range of both functions in interval notation:

11. Example 2 Sketch a graph of the function with the following characteristics: 1. Domain: (-8,-4) and Range: (-∞,∞) 2. Domain: [-2,3) and Range: (1,5)U[7,10]

12. Example 3 Find the domain and range of. t -32-20-155-40123 h 1087-7421ER DOMAIN:RANGE: The range is clear from the graph and table. The input to a square root function must be greater than or equal to 0 Dividing by a negative switches the sign

13. Slope Formula The slope of the line through the points (x 1, y 1 ) and (x 2, y 2 ) is given by:

14. Forms of a Line Point Slope Form - The equation of a line that contains the point (x 1,y 1 ) and whose slope is m is: Slope-Intercept Form - The equation of a line that contains the y-intercept (0,b) and whose slope is m is: General Form-

15. Parallel and Perpendicular Lines If the slope of line is then the slope of a line… Parallel is Perpendicular is

16. Example 1 Algebraically find the slope-intercept equation of a line that contains the points (-1,4) and (-4,-2). Find Slope 2 m (-1,4) (x 1,y 1 ) (-4,-2) (x 2,y 2 ) Substitute into point-slope

17. Example 2 Find an equation for the line that contains the point (2,-3) and is parallel to the line. Find the Slope of the original line: Find the equation of the Parallel line: Rewrite the equation into Slope- Intercept Form m We know a point and the slope Parallel lines have same slope

18. Basic Types of Transformations ( h, k ): The Key Point When negative, the original graph is flipped about the x-axis When negative, the original graph is flipped about the y-axis Horizontal shift of h units Vertical shift of k units Parent/Original Function: A vertical stretch if |a|>1and a vertical compression if |a|<1

19. Transformation Example Shift the parent graph four units to the left and three units down. Description: Use the graph of below to describe and sketch the graph of.

20. Piecewise Functions For Piecewise Functions, different formulas are used in different regions of the domain. Ex: An absolute value function can be written as a piecewise function:

21. Example 1 Write a piecewise function for each given graph.

22. Example 2 Rewrite as a piecewise function. Find the x value of the vertex Change the absolute values to parentheses. Plus make the one on the bottom negative. 4-4 6 x -3-201234 f(x)f(x) 65432123

23. Composition of Functions f g First Second OR Substituting a function or it’s value into another function. There are two notations: (inside parentheses always first)

24. Example 1 Let and. Find: Substitute x=1 into g(x) first Substitute the result into f(x) last

25. Example 2 Let and. Find: Substitute x into f(x) first Substitute the result into g(x) last