Functions and their Graphs

1. Functions and their Graphs

2. Relations • A relation is a mapping, or pairing, of input values with output values. • The set of input values is called the domain. • The set of output values is called the range. • A relation as a function provided there is exactly one output for each input. • It is NOT a function if at least one input has more than one output

3. In order for a relationship to be a function… Functions EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT (DOMAIN) FUNCTIONMACHINE (RANGE) OUTPUT

4. Example 6 Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

5. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

6. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

7. A Relation can be represented by a set of orderedpairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X<0, y>0 Origin (0,0) Quadrant IV X>0, y<0 Quadrant III X<0, y<0

8. Graphing Relations • To graph the relation in the previous example: • Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4) • Plot the points

9. (4,4) (-3,3) (1,1) (1,-2)

10. Vertical Line Test • You can use the vertical line test to visually determine if a relation is a function. • Slide any vertical line (pencil) across the graph to see if any two points lie on the same vertical line. • If there are not two points on the same vertical line then the relation is a function. • If there are two points on the same vertical line then the relation is NOT a function

11. Examples • I’m going to show you a series of graphs. • Determine whether or not these graphs are functions. • You do not need to draw the graphs in your notes.

12. YES! Function? #1

13. Function? #7 NO!

14. YES! Function? #2

15. Function? #5 NO!

16. Graphing and Evaluating Functions • Many functions can be represented by an equationin 2 variables: y=2x-7 • An ordered pair is a solution if the equation is true when the values of x & y are substituted into the equation. • Ex: (2,-3) is a solution of y=2x-7 because: • -3 = 2(2) – 7 • -3 = 4 – 7 • -3 = -3

17. In an equation, the input variable is called the independent variable. • The output variable is called the dependent variable and depends on the value of the input variable. • In y=2x-7 ….. X is the independent var. Y is the dependant var. • The graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation.

18. Graphing an equation in 2 variables • Construct a table of values • Graph enough solutions to recognize a pattern • Connect the points with a line or curve

19. Graph: y = x + 1 Step 3: Step2: Step 1 Table of values

20. Function Notation • By naming the function ‘f’ you can write the function notation: • f(x) = mx + b • “the value of f at x” • “f of x” • f(x) is another name for y (grown up name) • You can use other letters for f, like g or h

21. Decide if the function is linear. Then evaluate for x = -2 • f(x) = -x2 – 3x + 5 • Not linear…. • f(-2) = -(-2)2 – 3(-2) + 5 • f(-2) = 7 • g(x) = 2x + 6 • Is linear because x is to the first power • g(-2) = 2(-2) + 6 • g(-2) = 2 • The domain for both is….. • All reals

22. Assignment