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2.3: Functions

2.3: Functions. Objectives: To determine if a relation is a function To find the domain and range of a function To evaluate functions. Vocabulary. As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook. Relations.

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2.3: Functions

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  1. 2.3: Functions Objectives: • To determine if a relation is a function • To find the domain and range of a function • To evaluate functions

  2. Vocabulary • As a class, use your vast mathematical knowledge to define each of these words without the aid of your textbook.

  3. Relations A mathematical relation is the pairing up (mapping) of inputs and outputs.

  4. Relations A mathematical relation is the pairing up (mapping) of inputs and outputs. • Domain: the set of all input values • Range: the set of all output values

  5. Calvin and Hobbes! A toaster is an example of a function. You put in bread, the toaster performs a toasting function, and out pops toasted bread.

  6. Calvin and Hobbes! “What comes out of a toaster?” “It depends on what you put in.” • You can’t input bread and expect a waffle!

  7. What’s Your Function? A functionis a relation in which each input has exactly one output. • A function is a dependent relation • Output depends on the input Relations Functions

  8. What’s Your Function? A functionis a relation in which each input has exactly one output. • Each output does not necessarily have only one input Relations Functions

  9. How Many Girlfriends? If you think of the inputs as boys and the output as girls, then a function occurs when each boy has only one girlfriend. Otherwise the boy gets in BIG trouble. Darth Vadar as a “Procurer.”

  10. What’s a Function Look Like?

  11. What’s a Function Look Like?

  12. What’s a Function Look Like?

  13. What’s a Function Look Like?

  14. Exercise 1a Tell whether or not each table represents a function. Give the domain and range of each relationship.

  15. Exercise 1b The size of a set is called its cardinality. What must be true about the cardinalities of the domain and range of any function?

  16. Exercise 2 Which sets of ordered pairs represent functions? • {(1, 2), (2, 3), (3, 4), (3, 5), (5, 6)} • {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)} • {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1)} • {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5)}

  17. Break this Function!!!

  18. Exercise 3 Which of the following graphs represent functions? What is an easy way to tell that each input has only one output?

  19. Vertical Line Test A relation is a function iff no vertical line intersects the graph of the relation at more than one point If it does, then an input has more than one output. Function Not a Function

  20. Functions as Equations To determine if an equation represents a function, try solving the thing for y. • Make sure that there is only one value of y for every value of x.

  21. Exercise 4

  22. Set-Builder Notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: • {x | x < -2} reads “the set of all x such that x is less than negative 2”.

  23. Set-Builder Notation Since the domain or range of a function is often an infinite set of values, it is often convenient to represent your answers in set-builder notation. Examples: • {x : x < -2} reads “the set of all x such that x is less than negative 2”.

  24. Interval Notation Another way to describe an infinite set of numbers is with interval notation. • Parenthesis indicate that first or last number is notin the set: • Example: (-, -2) means the same thing as x < -2 • Neither the negative infinity or the negative 2 are included in the interval • Always write the smaller number, bigger number

  25. Interval Notation Another way to describe an infinite set of numbers is with interval notation. • Brackets indicate that first or last number is in the set: • Example: (-, -2] means the same thing as x  -2 • Infinity (positive or negative) never gets a bracket • Always write the smaller number, bigger number

  26. Domain and Range: Graphs • Domain: All x-values (L → R) • {x: -∞ < x < ∞} • Range: All y-values (D ↑ U) • {y: y ≥ -4} Range: Greater than or equal to -4 Domain: All real numbers

  27. Exercise 5 Determine the domain and range of each function.

  28. Domain and Range: Equations • Domain: What you are allowed to plug in for x. • Easier to ask what you can’t plug in for x. • Limited by division by zero or negative even roots • Can be explicit or implied • Range: What you can get out for y using the domain. • Easier to ask what you can’t get for y.

  29. Exercise 6a

  30. Exercise 6b

  31. Dependent Quantities Functions can also be thought of as dependent relationships. In a function, the value of the output depends on the value of the input. • Independent quantity: Input values, x-values, domain • Dependent quantity: Output value, which depends on the input value, y-values, range

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