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9/5/13 Warm Up

9/5/13 Warm Up. To get from A to B you must avoid walking through a pond by walking 34 m south and 41 m east. How many meters would be saved if it were possible to walk through the pond ? Write you answer t o the nearest meter (you may use a calculator)

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9/5/13 Warm Up

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  1. 9/5/13Warm Up • To get from A to B you must • avoid walking through a pond by • walking 34 m south and 41 m • east. How many meters would • be saved if it were possible to walk • through the pond? Write you answer • to the nearest meter (you may use a calculator) • 2. Find the distance between the two points (3, 4) and (-1, -2) • Write your answer in simplest radical form. 22 m 2√13

  2. Trig Game Plan Date: 9/5/13

  3. -7 -7 -2 -2 -1 -1 1 1 3 3 5 5 7 7 -6 -6 -5 -5 -4 -4 -3 -3 0 4 6 8 Inequality notation for graphs shown above. 2 Interval notation for graphs shown above. [ 0 2 4 6 8

  4. Squared bracket means can equal 4 Rounded bracket means cannotequal -2 -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Let's try another one. The brackets used in the interval notation above are the same ones used when you graph this. ( ] Set Notation: {x| -2 < x ≤ 4}

  5. Example 1 Write each set using interval notation. Create a graph (visual representation using a number line). • { x | x ≥ 5 } • “All values of x such that x is greater than or equal to 5.” • Answer: [5, ∞) [ 5

  6. Example 2 Write each set using interval notation. Create a graph (visual representation using a number line). • { x | -1 < x < 7 } • “All values of x such that x is greater than -1 but less than 7.” • Answer: (-1, 7) ( -1 ) 7

  7. Example 3 Write using interval notation and set notation Answer: (2, ∞) { x | x > 2 } (

  8. Example 4 Write using interval notation and set notation Answer: [-2, 1) { x | -2 ≤ x < 1} ) [

  9. Example 5 Write using interval notation and set notation Answer: (-∞, -4) U (4, ∞) { x | |x| > 4 } ) (

  10. Challenge Explain why the set { y | |y| ≥ 1 } is the same as (-∞, -1] U [1, ∞). Solution: After graphing the interval notation, we can see that the numbers between -1 and 1 are not solutions. [ ]

  11. Functions • For each value of x (domain) there is only one value of y (range). • “One-to-one” • Domain • X-axis • Independent • Input • Range • Y-axis • Dependent • Output

  12. HOW DO YOU KNOW IT’S A FUNCTION? • VERTICAL LINE (PENCIL) TEST • If every vertical line intersects the graph of a relation in no more than one point, then the graph is a function. • Are these functions? YES NO YES

  13. (b) (a) (d) (c) Which Are Functions? (a) and (c)

  14. Domain and Range

  15. The most common rules of algebra that limit the domain of functions are: Rule 1: You can’t divide by 0. Rule 2: You can’t take the square root of a negative number.

  16. Example 1 Deciding Whether Relations Define Functions • Decide whether the relation determines a function. (a) M is a function because each distinct x-value has exactly one y-value.

  17. Example 1 Deciding Whether Relations Define Functions (b) N is not a function because the x-value –4 has two y-values.

  18. Example 2(a) Finding Domains and Ranges of Relations (page 442) • Give the domain and range of the relation. Is the relation a function? {(–4, –2), (–1, 0), (1, 2), (3, 5)} Domain: {–4, –1, 0, 3} Range: {–2, 0, 2, 5} The relation is a function because each x-value corresponds to exactly one y-value.

  19. Example 2(b) Finding Domains and Ranges of Relations (cont.) • Give the domain and range of the relation. Is the relation a function? Domain: {1, 2, 3} Range: {4, 5, 6, 7} The relation is not a function because the x-value 2 corresponds to two y-values, 5 and 6.

  20. Example 3(a) Finding Domains and Ranges from Graphs (page 442) • Give the domain and range of the relation in set notation. Domain: {–2, 4} Range: {0, 3}

  21. Domain: Range: Example 3(b) Finding Domains and Ranges from Graphs • Give the domain and range of the relation in interval notation.

  22. Example 3(c) Finding Domains and Ranges from Graphs • Give the domain and range of the relation in interval notation. Domain: [–5, 5] Range: [–3, 3]

  23. Domain: Range: Example 3(d) Finding Domains and Ranges from Graphs • Give the domain and range of the relation in interval notation.

  24. Domain: Range: Example 4(a) Identifying Function Domains, and Ranges • Determine if the relation is a function and give the domain and range. y = 2x– 5 y is found by multiplying x by 2 and subtracting 5. Each value of x corresponds to just one value of y, so the relation is a function.

  25. Domain: Range: Example 4(b) Identifying Functions, Domains, & Ranges (cont.) • Determine if the relation is a function and give the domain and range. y = x2+ 3 y is found by squaring x by 2 and adding 3. Each value of x corresponds to just one value of y, so the relation is a function.

  26. Domain: Range: Example 4(c) Identifying Functions, Domains, & Ranges (cont.) • Determine if the relation is a function and give the domain and range. x = |y| For any choice of x in the domain, there are two possible values for y. The relation is not a function.

  27. Domain: Range: Example 4(d) Identifying Functions, Domains, and Ranges (cont.) • Determine if the relation is a function and give the domain and range. y is found by dividing 3 by x + 2. Each value of x corresponds to just one value of y, so the relation is a function.

  28. Example #5 RANGE Find the domain and range in interval notation. Determine if the following is a function. DOMAIN D: [-3, 7] R: [-8, 2] Not it’s not a function.

  29. RANGE Example #6 DOMAIN Find the domain and range in interval notation. Determine if the following is a function. D: (-∞, ∞) R: [2, ∞) It’s a function.

  30. What is the domain and range of the following relation? Is this a function? Why or why not? { (-1,2), (2, 51), (1, 3), (8, 22), (9, 51) } Domain: -1, 2, 1, 8, 9 Range:2, 51, 3, 22, 51 Function: Yes, no domain (x) values repeat. It’s one-to-one. Passes vertical line test. For the following relation to be a function, X can not be what values? { (8, 11), (34,5), (6,17), (X ,22) } X cannot be 8, 34, or 6.

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