Module 2 topic 1 notes

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Module 2 topic 1 notes

  1. 1. Module 2 Topic 1
  2. 2. <ul><li>In Algebra I you learned about relations and functions. You will recall that a relation is a set of ordered pairs, and a function is a special relation that assigns each value in the domain (the first coordinate in the ordered pair, or the x value) to exactly one value in the range (the second coordinate in the ordered pair, or the y value). </li></ul>Let’s take a minute and do a more thorough review of these concepts. Go now to http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=133 , login, and launch the Gizmo Introduction to Functions. Open the Exploration Guide and follow the instructions.
  3. 3. Practice <ul><li>Try these problems on your own. Once you have answered a problem, click once to see the answer. Remember to work each problem first, then click to see the answer. </li></ul><ul><li>Use a mapping diagram to determine if the relation is a function </li></ul><ul><ul><li>(2, 7), (4, 1), (5, 9), (3, 2), (8, 0) </li></ul></ul>2 3 4 5 8 0 1 2 7 9 Since each value in the domain matches to only one value in the range, the relation is a function.
  4. 4. <ul><li>Use a mapping diagram to determine if the relation is a function. </li></ul><ul><li>Use a mapping diagram to determine if the relation is a function. </li></ul><ul><ul><li>(8, 1), (3, 4), (2, 7), (8, 5), (0, 6) </li></ul></ul>domain range 1 2 3 5 9 3 4 5 9 Since each value in the domain matches to only one value in the range, the relation is a function. domain range 0 2 3 8 1 4 5 6 7 Since a value in the domain (8) matches more than one value in the range (1 and 5), the relation is not a function. 2 5 3 1 9 5 9 4 4 3
  5. 5. <ul><li>4. Use the vertical line test to determine if the relation is a function. </li></ul>Since a vertical line never touches more than one point at a time, the relation is a function.
  6. 6. Evaluating functions <ul><li>Click on the link to see notes on evaluating functions. </li></ul><ul><li>www.regentsprep.org/Regents/math/algtrig/ATP5/Evaluating </li></ul>
  7. 7. Practice Try these problems on your own. Once you have answered a problem, click once to see the answer. Remember to work each problem first, then click to see the answer. 3(7) – 4 21 – 4 17 5 2 + 2(5) + 1 25 + 10 + 1 36 f(x) = 3x – 4 for x = 7 f(x) = x 2 + 2x + 1 for x = 5
  8. 8. Practice <ul><li>Find the range for the function for the domain </li></ul><ul><li>{4, 8, 12, 20, 32} </li></ul><ul><li>Range = {1, 2, 3, 5, 8} </li></ul>
  9. 9. In Algebra 1 you learned the basics of domain and range. Click on the link below to watch a video and refresh your memory on that topic. Pearson Prentice Hall Mathematics Video
  10. 10. Many functions can be represented by an equation in two variables. The input variable is called the independent variable. The output variable in an equation, which depends on the value of the input variable, is called the dependent variable. The domain of a function consists of the set of all input values. The range of a function consist of the set of all output values. A function is like a machine that you input numbers and variables. The machine alters the input in some way and produces an answer.
  11. 11. Example: Find the domain and range of the following graph: What's the domain ? The graph above is represented by y = x 2 , and we can square any number we want. Therefore, the domain is all real numbers. On a graph the domain corresponds to the horizontal axis.  Since that is the case, we need to look to the left and right to see if there are any end points or holes in the graph to help us find our domain. If the graph keeps going on and on to the right and the graph keeps going on and on to the left then the domain is represented by all real numbers.
  12. 12. What's the range ? If I plug any number into this function, am I ever going to be able to get a negative number out of it? No, not in the Real Number System! The range of this function is all positive numbers which is represented by y ≥ 0. On a graph, the range corresponds to the vertical axis.  Since that is the case, we need to look up and down to see if there are any end points or holes to help us find our range. If the graph keeps going up and down with no endpoint then the range is all real numbers. However, this is not the case. The graph does not begin to touch the y-axis until x = 0, then it continues up with no endpoints which is represented by y ≥ 0.
  13. 13. Let's try another example: What numbers can we plug into this function? What happens if we plug in 4? If x = 4, we divide by zero which is undefined. Therefore, the domain of this function is: all real numbers except 4. The range is all real numbers except 0 . (We can only produce zero when a zero is in the numerator.) , zero in the numerator is OK! However, , zero in the denominator is a NO NO! In general, when you're trying to find the domain of a function, there are two things you should look out for. (1) Look for potential division by zero. (2) Look for places where you might take the square root of a negative number. (3) If you have a verbal model, you can only use numbers that make sense in the given situation. (This is called the relevant domain as it only identifies the values that make sense in the given situation.)
  14. 14. Practice Find the domain and range a. {(2, 8), (5, 7), (6, 9), (3, 4)} b. <ul><li>Domain = {2, 3, 5, 6} </li></ul><ul><li>Range = {4, 7, 8, 9} </li></ul><ul><li>Domain = all real numbers </li></ul><ul><li>range: y ≥ 2 </li></ul>
  15. 15. Summary The domain of a function is all the possible input values, and the range is all possible output values.

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