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# Functions

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### Functions

1. 1. Functions Mathematics 4 June 20, 2012Mathematics 4 () Functions June 20, 2012 1/1
2. 2. DeﬁnitionsRelationsRelationsA set of ordered pairs (x, y) such that for each x-value, there correspondsat least one y-value. Mathematics 4 () Functions June 20, 2012 2/1
3. 3. DeﬁnitionsFunctionFunctionA set of ordered pairs (x, y) such that for each x-value, there correspondsexactly one y-value.FunctionA correspondence from a set X ⊆ R to a set Y ⊆ R where x ∈ X andy ∈ Y , and y is unique for a speciﬁc value of x. Mathematics 4 () Functions June 20, 2012 3/1
4. 4. DeﬁnitionsOne-to-one Function Mathematics 4 () Functions June 20, 2012 4/1
5. 5. DeﬁnitionsDomain and RangeDomainThe domain of a function is the set of all possible values of x(independent variable, abscissa) for a given relation or function.RangeThe range of a function is the set of all possible values of y (dependentvariable, ordinate) for a given relation or function. Mathematics 4 () Functions June 20, 2012 5/1
6. 6. ExamplesDeﬁnitions of Functions, Domain and RangeIdentify if the following relations are functions, and give the domainand range. 1 y = x2 + 6x + 4 Mathematics 4 () Functions June 20, 2012 6/1
7. 7. ExamplesDeﬁnitions of Functions, Domain and RangeIdentify if the following relations are functions, and give the domainand range. 1 y = x2 + 6x + 4 2 x2 + y 2 = 1 Mathematics 4 () Functions June 20, 2012 6/1
8. 8. ExamplesDeﬁnitions of Functions, Domain and RangeIdentify if the following relations are functions, and give the domainand range. 1 y = x2 + 6x + 4 2 x2 + y 2 = 1 1 3 y= x+1 Mathematics 4 () Functions June 20, 2012 6/1
9. 9. ExamplesDeﬁnitions of Functions, Domain and RangeIdentify if the following relations are functions, and give the domainand range. 1 y = x2 + 6x + 4 2 x2 + y 2 = 1 1 3 y= x+1 1 4 y= x2 +1 Mathematics 4 () Functions June 20, 2012 6/1
10. 10. Homework 5Identify if the following relations are functions, and give the domain and range. 3 1 y= x+1 3x2 + 1 2 y= x2 + 2 √ 3 y = −x2 + 25 4 x2 + y 2 = 100 5 y + 3 = (x + 4)2 2 6 y= |x| Mathematics 4 () Functions June 20, 2012 7/1
11. 11. Function Notation Given the equation y = 2x2 + 5 Mathematics 4 () Functions June 20, 2012 8/1
12. 12. Function Notation Given the equation y = 2x2 + 5 Using the set-builder notation and the deﬁnition of functions: f = {(x, y) y = 2x2 + 5 } Mathematics 4 () Functions June 20, 2012 8/1
13. 13. Function Notation Given the equation y = 2x2 + 5 Using the set-builder notation and the deﬁnition of functions: f = {(x, y) y = 2x2 + 5 } From this notation we can use the shorthand: f (x) = 2x2 + 5 Mathematics 4 () Functions June 20, 2012 8/1
14. 14. DeﬁnitionsGraphs of FunctionsThe graph of a functionThe graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function. Mathematics 4 () Functions June 20, 2012 9/1
15. 15. DeﬁnitionsGraphs of FunctionsThe graph of a functionThe graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function.Vertical Line TestThe graph of a function can be intersected by a vertical line in at mostone point. Mathematics 4 () Functions June 20, 2012 9/1
16. 16. Example:Square root functions √Find the graph of the function f = {(x, y) y = 4 − x }: Mathematics 4 () Functions June 20, 2012 10 / 1
17. 17. Example:Square root functions √Find the graph of the function f = {(x, y) y = x − 1 }: Mathematics 4 () Functions June 20, 2012 11 / 1
18. 18. Example:Absolute value functionsFind the graph of the function f = {(x, y) |y = |x − 3| }: Mathematics 4 () Functions June 20, 2012 12 / 1
19. 19. Homework 6Sketch the graph and determine domain and range for each function below. √ 1 f = {(x, y) | y = 16 − x2 } 2 g = {(x, y) | y = (x − 1)3 } x2 − 4x + 3 3 h= (x, y) | y = x−1 Mathematics 4 () Functions June 20, 2012 13 / 1
20. 20. Evaluating FunctionsEvaluating FunctionsAssign values to the independent variable and simplifying.Evaluate the following: f (x + h) − f (x) 1 if f (x) = 3x2 − 2x + 4 h Mathematics 4 () Functions June 20, 2012 14 / 1
21. 21. Evaluating FunctionsEvaluating FunctionsAssign values to the independent variable and simplifying.Evaluate the following: f (x + h) − f (x) 1 if f (x) = 3x2 − 2x + 4 h 2 f (−x) if f (x) = 3x4 − 2x2 + 7 Mathematics 4 () Functions June 20, 2012 14 / 1
22. 22. Evaluating FunctionsEvaluating FunctionsAssign values to the independent variable and simplifying.Evaluate the following: f (x + h) − f (x) 1 if f (x) = 3x2 − 2x + 4 h 2 f (−x) if f (x) = 3x4 − 2x2 + 7 3 g(−x) if g(x) = 3x5 − 4x3 − 9x Mathematics 4 () Functions June 20, 2012 14 / 1
23. 23. Evaluating FunctionsEvaluating FunctionsAssign values to the independent variable and simplifying.Evaluate the following: f (x + h) − f (x) 1 if f (x) = 3x2 − 2x + 4 h 2 f (−x) if f (x) = 3x4 − 2x2 + 7 3 g(−x) if g(x) = 3x5 − 4x3 − 9x 1+x 4 f (x2 − 1) if f (x) = 2x − 1 Mathematics 4 () Functions June 20, 2012 14 / 1
24. 24. Operations on FunctionsThe following notations are indicate an operation between two functions: (f + g)(x) = f (x) + g(x) (f − g)(x) = f (x) − g(x) (f · g)(x) = f (x) · g(x) f f (x) (x) = g g(x) Mathematics 4 () Functions June 20, 2012 15 / 1
25. 25. Operations on FunctionsDetermine the result of the following function operations: x+3 1 (f + g)(x) if f (x) = and g(x) = x − 2 x+2 Mathematics 4 () Functions June 20, 2012 16 / 1
26. 26. Operations on FunctionsDetermine the result of the following function operations: x+3 1 (f + g)(x) if f (x) = and g(x) = x − 2 x+2 f √ √ 2 (x) if f (x) = x3 − x2 − 5x − 3 and g(x) = x−3 g Mathematics 4 () Functions June 20, 2012 16 / 1
27. 27. Composition of FunctionsDeﬁnitionComposition of FunctionsEvaluating a function f (x) with another function g(x).f (g(x)) = (f ◦ g)(x) Mathematics 4 () Functions June 20, 2012 17 / 1
28. 28. Composition of FunctionsEvaluate the following composite functions: x+1 1 1 f (x) = , g(x) = , ﬁnd (f ◦ g) and (g ◦ f ) x−1 x Mathematics 4 () Functions June 20, 2012 18 / 1
29. 29. Composition of FunctionsEvaluate the following composite functions: x+1 1 1 f (x) = , g(x) = , ﬁnd (f ◦ g) and (g ◦ f ) x−1 x √ √ 2 f (x) = x2 − 1, g(x) = x − 1, ﬁnd (f ◦ g)(x) and (g ◦ f ) Mathematics 4 () Functions June 20, 2012 18 / 1
30. 30. Odd and Even FunctionsDeﬁnitionsEven FunctionA function f is even if f (−x) = f (x).Example 1 f (x) = 3x6 − 2x4 + 4x2 + 2 Mathematics 4 () Functions June 20, 2012 19 / 1
31. 31. Odd and Even FunctionsDeﬁnitionsEven FunctionA function f is even if f (−x) = f (x).Example 1 f (x) = 3x6 − 2x4 + 4x2 + 2 2 g(x) = |x| + 2 Mathematics 4 () Functions June 20, 2012 19 / 1
32. 32. Odd and Even FunctionsDeﬁnitionsOdd FunctionA function f is odd if f (−x) = −f (x).Example 1 f (x) = 2x5 − 4x3 + 5x Mathematics 4 () Functions June 20, 2012 20 / 1
33. 33. Odd and Even FunctionsDeﬁnitionsOdd FunctionA function f is odd if f (−x) = −f (x).Example 1 f (x) = 2x5 − 4x3 + 5x 1 2 g(x) = x Mathematics 4 () Functions June 20, 2012 20 / 1
34. 34. Odd and Even FunctionsDetermine if the following functions are odd, even, or neither (1) (2) Mathematics 4 () Functions June 20, 2012 21 / 1
35. 35. Odd and Even FunctionsDetermine if the following functions are odd, even, or neither (3) (4) Mathematics 4 () Functions June 20, 2012 22 / 1
36. 36. Odd and Even FunctionsDetermine if the following functions are odd, even, or neither (5) (6) Mathematics 4 () Functions June 20, 2012 23 / 1
37. 37. Odd and Even FunctionsDetermine if the following functions are odd, even, or neither (7) (8) Mathematics 4 () Functions June 20, 2012 24 / 1
38. 38. Odd and Even FunctionsSymmetry propertiesEven functionsThe graph of even functions are symmetric with respect to the y-axis. Mathematics 4 () Functions June 20, 2012 25 / 1
39. 39. Odd and Even FunctionsSymmetry propertiesEven functionsThe graph of even functions are symmetric with respect to the y-axis.Odd functionsThe graph of odd functions are symmetric with respect to the origin. Mathematics 4 () Functions June 20, 2012 25 / 1
40. 40. Homework 7Determine if the function is odd/even/neither, then ﬁnd the domain, range, and thegraph of the function. x 1 f (x) = x2 −4 2 (f + g)(x) if f (x) = x2 + 1 and g(x) = |x| 3 (f − g)(x) if f (x) = |x| + 1 and g(x) = x2 − 2 4 (f · g) if f (x) = x and g(x) = x3 Mathematics 4 () Functions June 20, 2012 26 / 1