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Understanding Functions and Relations in Mathematics

Explore the concepts of functions and relations as correspondences between sets, from domain to range, with examples and equations explained. Dive into linear functions, graphing functions, and evaluating functions for practical applications like falling objects.

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Understanding Functions and Relations in Mathematics

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  1. Functions

  2. Relation A relation is a correspondence between two sets where each element in the first set, called the domain, corresponds to at least one element in the second set, called the range.

  3. Relation

  4. Relation

  5. Relation The domain is the set of all the first components. {Michael, Tania, Dylan, Trevor, Megan} The range is the set of all the second components. {A, AB, O}

  6. Relation The domain is the set of all the first components. {Michael, Tania, Dylan, Trevor, Megan} The range is the set of all the second components. {A, AB, O}

  7. Function

  8. Function • A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactlyone element in the second set, called the range

  9. Function • A function is a correspondence between two sets where each element in the first set, called the domain, corresponds to exactlyone element in the second set, called the range • Note that the definition of a function is more restrictive than the definition of a relation.

  10. Function

  11. Functions Defined by Equations

  12. Functions Defined by Equations y = x2 − 3x

  13. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y

  14. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1

  15. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1)

  16. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2

  17. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5

  18. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5 y = (5)2 − 3(5)

  19. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5 y = (5)2 − 3(5) 10

  20. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5 y = (5)2 − 3(5) 10 1.2

  21. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5 y = (5)2 − 3(5) 10 1.2 y = (1.2)2 − 3(1.2)

  22. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y  1 y = (1)2 − 3(1) −2 5 y = (5)2 − 3(5) 10 1.2 y = (1.2)2 − 3(1.2) −2.16

  23. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y •  Since the variable y depends on what value of x is selected, we denote y as the dependent variable. (output)

  24. Functions Defined by Equations y = x2 − 3x x y = x2 − 3x y •  Since the variable y depends on what value of x is selected, we denote y as the dependent variable. (output) • The variable x can be any number in the domain; therefore, we denote x as the independent variable. (input)

  25. Function Notation

  26. Function Notation • The notation y = f(x) denotes that the variable y is function of x.

  27. Function Notation • The notation y = f(x) denotes that the variable y is function of x. INPUT FUNCTION OUTPUT EQUATION x f f (x) f (x) = 2x + 5

  28. Function Notation • A Linear function is a function defined by an equation that can be written in the form f(x) = mx + b , or y = mx + b where m is the slope of the line graph and (0, b) is the y - intercept

  29. Function Notation • A Linear function is a function defined by an equation that can be written in the form f(x) = mx + b , or y = mx + b where m is the slope of the line graph and (0, b) is the y - intercept Ex. y = -3x + 8 f(x) = 5x – 4

  30. The Graph of the Function • The graph of the function is the graph of the ordered pairs (x, f(x)), that define the function.

  31. Use the given graphs to evaluate the function. Find f (0), f (1). f (2) , 4f (3), Find x such that f (x) = 10, f (x) = 2 Find x such that f (x) = 10, f (x) = 2

  32. Use the given graphs to evaluate the function. T(−5) T(−2) T(4)

  33. Vertical Line Test • Given the graph of an equation, if any vertical line that can be drawn intersects the graph at no more than one point, the equation defines a function of x. This test is called the vertical line test.

  34. Vertical Line Test

  35. Evaluating the Difference Quotient

  36. Evaluating the Difference Quotient

  37. Evaluating the Difference Quotient For the function f (x) = x2 − x, find

  38. Falling Objects: Firecrackers. • A firecracker is launched straight up, and its height is a function of time, h(t) = −16t2 + 128t, where h is the height in feet and t is the time in seconds with t = 0 corresponding to the instant it launches. What is the height 4 seconds after launch?

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