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Module 1 Lesson 1 Remediation Notes

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Module 1 Lesson 1 Remediation Notes

1. 1. Remediation Notes
2. 2. Relation Function Every equation/graph/set of Functions are just relations ordered pairs represents a relation, but sometimes a relation is a function. in which the x values of its points (ordered pairs) do not repeat. If a graph passes the vertical line test, then it is the graph of a function.
3. 3. To determine if a graph is a function, we use the vertical line test. If it passes the vertical line test then it is a function. If it does not pass the vertical line test then it is not a function.
4. 4. Vertical Line Test: 1.Draw a vertical line through the graph. 2. See how many times the vertical line intersects the graph at any one location. If Only Once – Pass (function) If More than Once – Fail (not function)
5. 5. Is this graph a function? Only crosses at one point. Yes, this is a function because it passes the vertical line test.
6. 6. Is this graph a function? Crosses at more than one point. No, this is not a function because it does not pass the vertical line test.
7. 7. To determine if a table represents a function, we look at the x column (domain). If each number in the x column appears only once in that column, it is a function.
8. 8. Relations and Functions You can use the vertical line test to determine whether a relation is a function. Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. If some vertical line intercepts a graph in two or more points, the graph does not represent a function.
9. 9. Relations and Functions State the domain and range of the relation shown in the graph. Is the relation a function? y (-4,3) (2,3) The relation is: { (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) } The domain is: { -4, -1, 0, 2, 3 } x (-1,-2) The range is: { -4, -3, -2, 3 } (0,-4) Each member of the domain is paired with exactly one member of the range, so this relation is a function. (3,-3)
10. 10. Is this relation a function? X 1 2 3 4 Y 5 6 5 8 Every number just appears once. Yes, this is a function because each number in the x column only appears once.
11. 11. Is this relation a function? X 24 6 10 10 Y 7 9 8 10 The number 10 appears more than once. No, this is not a function because 10 appears in the x column more than once.
12. 12. To Evaluate a Function for f(#): Plug the # given in the (#) into all x’s Simplify Try these… http://www.mathslideshow.com/Alg2/Lesson2-1/fv4.htm
13. 13. Functions Remember f(x), g(x), h(x), … all just mean y. We use f(x), g(x), h(x), … when we have more than one y = equation.
14. 14. Review Evaluate f ( x) = x 2 − 2 x + 5 for f (3). f(3) = (3)2 – 2(3) + 5 f(3) = 8 Evaluate f ( x) = 5 x 3 − 2 x − 8 for f (−1). f(-1) = 5(-1)3 – 2(-1) – 8 f(-1) = -11
15. 15. Basic function operations Sum ( f + g) ( x) = f ( x) + g ( x) Difference ( f – g) ( x) = f ( x) – g ( x) Product ( f × g )( x ) = Quotient f ( x) f  , g ( x) ≠ 0 ( f g) ( x) =  ( x) = g ( x) g ©1999 by Design Science, Inc. 15 f ( x) × g ( x)
16. 16. f ( x) = 2 x + 3 g ( x) = −5 x − 9 f ( x) − g ( x) = (2 x + 3) − (−5 x − 9) You MUST DISTRIBUTE the NEGATIVE f ( x) − g ( x) = 2 x + 3 + 5 x + 9 f ( x) − g ( x) = 7 x + 12
17. 17. g ( x) = −5 x − 9 f ( x) = 2 x + 3 f ( x) • g ( x) = ( 2 x + 3)( − 5 x − 9 ) You MUST f ( x) • g ( x) = −10 x 2 − 18 x − 15 x − 27 f ( x) • g ( x) = −10 x − 33 x − 27 2 FOIL
18. 18. Domain and Range: If you are given a set of ordered pairs or a graph (which you would find the ordered pairs all by yourself) The x values are the DOMAIN The y values are the RANGE { (-3,5) , (-1, 6), (0, 4), (2, 3.5), (6, 13), (6, 29} Domain: { -3, -1, 0, 2, 6 } Range: { 3.5, 4, 5, 6, 13, 29}
19. 19. Domain and Range: If the equation is a line (y = mx + b or y = #) DOMAIN AND RANGE ARE ALL REAL NUMBERS ALWAYS!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!
20. 20. Domain and Range: If there is an x in the denominator of a fraction, you need to find the value of x that makes the ENTIRE DENOMINATOR equal zero. This number is the EXCEPTION to the DOMAIN of all real numbers. could be anything x Domain is all real numbers except 0 could be anything x −5 Domain is all real numbers except 5 could be anything x+9 Domain is all real numbers except -9
21. 21. Domain and Range: If you are given a line segment The DOMAIN (x values) is written like #< x < # The RANGE (y values) is written like #< y < # #< x < # #< y < #
22. 22. Domain and Range: If you are given a parabola The DOMAIN is ALWAYS ALL REAL NUMBERS The RANGE (y values) is written like y > # or y< #
23. 23. Find domain and range from an equation Most of the functions you study in this course will have all real numbers for both the domain and range. We’ll only look at the domain for exceptions: 1. Fractions: cannot have the denominator (bottom) = 0, so domain cannot be any x-value that makes the denominator= 0 Examples 3 f ( x) = x y= Domain: x≠0 f ( x) = x x−3 Domain: x≠3 (it’s okay for x=0 on top!) x2 +1 x 2 − 1 Domain: x≠1 or -1 because they both make the denominator=0 Question: How can you calculate which values make the denominator = 0? Set up the equation denominator = 0 and solve it. Those values are NOT allowed!
24. 24. Review
25. 25. Examples y 5 ● ● ● -5 -5 y 5 5 Domain: {-3,-2,1,3} Range: {0, -3} *Don’t repeat y Domain: y ≥ 0 Range: *Graph continues rt x x x ● x≥3 y 5 5 5 5 Domain: − 2 ≤ x ≤ 1 Range: y=4 or {4} *x is between -2 and 1 Domain: x is any real # Range: y ≤ 2 *Graph continues down 5 5 Domain: {x| − 5 4≤ x≤ 3} Range: {y| − 4 ≤ y ≤ 3 } *This is “set notation” Domain: x is any real # Range: y is any real # *Graph continues all ways
26. 26. Does the graph represent a function? Name the domain and range. x y x y Yes D: all reals R: all reals Yes D: all reals R: y ≥ -6
27. 27. Does the graph represent a function? Name the domain and range. x No D: x ≥ 1/2 R: all reals y x y No D: all reals R: all reals
28. 28. Visit these sites for remediation: http://www.purplemath.com/modules/fcnops.htm http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col _alg_tut30b_operations.htm http://teachers.henrico.k12.va.us/math/hcpsalgebra2/2-1.htm