1. Understanding the Relationships of Functions & Systems of Equations MATH 2 RICHARDSON 423 2/16/15

2. Week overview This week’s lesson entails a further look into polynomials and their behavior in a graphical application. This week as a class we will be able to identify the following: Review of a Cartesian graph Vocabulary Terms A function Definition Identifying examples Class of functiona Degree of functions Mathematical Terms Practice Examples

3. Work overview continued Equations Definition Examples of Different Equations Terms Practice Examples Inequalities Definition Delineation of equations and inequalities Examples of various cases by graphing problems/solution

4. Parts of a Cartesian (Plane)graph Vocabulary terms: Axis: Y-axis Vertical directed line X-axis Horizontal directed line that reads for all points perpendicular to the y axis. Coordinate: Points along axis meniscus found from solutions created from selected functions/equations Origin The center point of the x-axis and y-axis.a

5. Cartesian (Plane) Graph Illustration

6. The Coordinates Illustration As you can see from the illustration, a pair of coordinates Are made by moving either left to right along the x-axis first. (X, The y values are found by locating the values either up or down parallel to the y-axis.,Y)

7. Understanding what is a function? What exactly is a function? Defitinition: A function is a special relationship where each input has a single output. A function is often written as “f(x)” where the x is the input value. Example: F(x) =( x/3) (Orally: “F of x is(=) x divided by 3” This is a function because each input “X” has a single output “x/3” F(3)= ( 3/3)=1 F(15)= (15/3)=5 F(-12)=(-12/3)=-4

8. Let’s understand the three parts of a function The inputThe relationshipThe Output The value to place into the function The actual equation we are substituting the value for x The final solution once substitution has been made We address the input as “F( )” x 2 +3x+9= ______________ F(0)(0) 2 +3(0)+9=9 F(1)(1) 2 +3(1)+9=13

9. Function Rules Functions are just equations we just substitute values in to find solutions. Functions follow this saying: “Not all functions are equations, but rather mathematical relationships.” Meaning they don’t have to follow the same rules or mathematical applications as equations. Abstractly an equation is a lightly defined statement with some variables that can lead to a definite solution to multiple answers. A function relates only to one variable directly that will result to a set of solutions directly equal. A One to one description of a mathematical relationship of numbers.

10. Functions follow these rules: 1.“…each element….” 1.A function relates each element of a set with exactly one element of another set (possibly the same set). 2. “….exactly one…” A functioin is single valued. It will not give you back 2 or more results for the same input. Example: f(2)= 7 OR 9

11. Vertical Line Test So after we find our output values and plot them on our Cartesian plane how are we sure that we are dealing with an actual function instead of a mathematical relationship? We use a vertical line test. On a graph, the idea of a single valued means that no vertical line ever crosses more that one value in passing across the final ‘coordinate pairs’. If it crosses more than once it is a still a valid curve that describes the results, BUT IT IS NOT A FUNCTION.

12. Practice Examples on Understanding Functions Write an equation to represent the function from the following table of values: XY -2-4 -2 00 12 24 A. Y=-2XB. Y=2X C. Y=X+1D. Y=X+2

13. Practice Example Which one of the following relations is NOT a function?

14. Which one of these graphs does not illustrate a function? Hint: Use the Vertical Line Test to solve this problem.

15. Which one of the following graphs is not a function?

16. UNDERSTANDING DOMAIN AND RANGE WHAT IS A DOMAIN? Outside of the terminology for cyberspace pertaining to an identification string which constitutes a brand and space for a product by way of html code,a domain is a serious term we use in math to define elements. The DOMAIN is a set of all the values used to go into a function. These would be the values located on the “ X-AXIS” The RANGE is the output values made from the function. The output values or solutions from the function would be the values located by the “Y-AXIS”

17. Domain and Range Illustration

18. Practice Example A = {-3, -2, -1, 0, 1, 2, 3} f is a function from A to the set of whole numbers as defined in the following table: A. The set of IntegersB. The set of whole numbers C. {-3,-2,-1,0,1,2,3}D: {0,1,4,9}

19. Practice Example Which relation is not a function? A. F(x)=√xB. f=-√x C. F(x)=+√xD. F(x)=√x -1

20. Practice Example The function f is defined on the real numbers by f(x) =2+x-x 2 ? What is the value of f(-3)? A. -10B. -4 C. 8D. 14