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College Algebra: Functions Activity - Sponsored by ACEE and NSF

Learn about functions in college algebra through this sponsored activity. Understand the definition of a function and how to determine if a graph represents a function using the vertical line test. Practice finding the domain of various functions.

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College Algebra: Functions Activity - Sponsored by ACEE and NSF

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  1. COLLEGE ALGEBRA Sponsored in Part by ACEE and NSF By Vicki Norwich and Jacci White

  2. Activity 1 FUNCTIONS

  3. FUNCTIONS • A function is a rule that assigns a single output to each input. • Definition: A relation that assigns to each member of its domain exactly one member, its range. • Vertical line test: If it is possible for a vertical line to intersect a graph more than once, the graph is not the graph of a function. Functions

  4. This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time. Functions

  5. This example is not a function because if you draw a vertical line anywhere near the middle of the graph, it will cross more than one time. Functions

  6. Practice: Is this the graph of a function? Functions

  7. This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time. Functions

  8. Practice: Is this the graph of a function? Functions

  9. This example is not a function because if you draw a vertical line anywhere, it will cross the graph more than one time. near the middle of Functions

  10. Practice: Is this the graph of a function? Functions

  11. This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time. Functions

  12. Practice: Is this the graph of a function? Functions

  13. This example is a function because no matter where you draw a vertical line it crosses the graph no more than 1 time. Functions

  14. Practice: Is this the graph of a function? Functions

  15. This example is not a function because if you draw a vertical line anywhere on the right side of the graph, it will cross more than one time. Functions

  16. DOMAIN • The Denominator cannot equal zero. So the domain is made up of all real numbers that will not make the denominator equal to zero Functions

  17. The domain of this function is the set of all real numbers not equal to 3. Examples Functions

  18. What is the domain of f(x)? Functions

  19. The domain of the prior function is the set of all real numbers not equal to 2. Answer Functions

  20. Another way to write the answer is: Functions

  21. Another example of a function and the domain. Functions

  22. Practice: What is the domain of the following function? Functions

  23. Your answers should be: • The set of all real numbers not equal to 0,4, and -3. Functions

  24. Domain • An even index must have a radicand greater than or equal to zero. In other words, you cannot take an even root of a negative number. Functions

  25. Examples • The following function is a square root function. Because square root is even, the part of the function under the square root sign must be greater than or equal to zero. Functions

  26. To find the domain of a function that has an even index, set the part under the radical greater than or equal to 0 and solve for x. Therefore, the answer is all real numbers x>4 Solution

  27. Practice: What is the domain of the following function? Functions

  28. Your answers should be: Functions

  29. All rules for domains must be used whenever they apply. • What is the domain of f(x)=x-2? • What is the domain of g(x)=4x+7 • What is the domain of s(t)=(x+5)(x-2)? Functions

  30. Solution: All real numbers The solution is all real numbers for each of the prior three examples because there are no denominators and no radicals.

  31. Practice: What are the domains of the following functions? 1. 2. 3. Functions

  32. Your answers should be: 1. 2. All real numbers 3. Functions

  33. Reason for the last three answers. • In the first problem you have to factor the denominator to see when it will equal zero. • The second function has no fractions (denominator) and no radicals so the answer is all real numbers. • In the last problem you must set the 3x that is under the radical sign greater than or equal to zero and solve for x by dividing by 3 on both sides.

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