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Functions By Adrienne Calomino
<ul><li>Objectives: </li></ul><ul><li>to identify the domain and range of a relation - review; </li></ul><ul><li>to recogn...
x or input value y or output value <ul><li>Review domain and range of a relation </li></ul><ul><ul><li>Warm–up exercises <...
<ul><li>When is a relation a function? </li></ul><ul><li>When each element in the domain maps to one and only one element ...
<ul><li>Is this a Function?  Why or why not? </li></ul>Dog Cat Duck Lion Pig Rabbit 11 10 7 Domain Range
<ul><li>How about this? </li></ul>Dog Cat Duck Lion Pig Rabbit 11 9 10 6 7 5 Domain Range <ul><li>This is a particular typ...
Function <ul><ul><li>A function maps each element of the domain to  </li></ul></ul><ul><ul><li>one and only one element of...
Which is a function? Why? Table A Table B Table A Range Domain 7 5 6 4 5 4 5 3 3 2 3 1 2 1 output input Range Domain 8 7 7...
Which is a function? Why? Table A Table B Table A 120 Russian 95 Japan 48 Pakistan 48 France 72 Italy 72 Germany 47 Mexico...
<ul><li>Try coordinate points or ordered pairs </li></ul>Which is a function? Why? { (0,1) (0,-1), (1,2), (2,1), (3,1), (4...
<ul><li>1) Now try some from your worksheet </li></ul><ul><li>2) Group activity </li></ul><ul><ul><li>Make up two problems...
<ul><li>Definition </li></ul><ul><li>We say “function”.  We write  f. </li></ul><ul><li>We say “f of x”.  We write  f(x). ...
Mapping a function
Why are these functions?  Give an example of an argument of a function. What is the domain and range? Table A Table B f(0)...
<ul><li>Now let’s relate coordinate points to a graph. </li></ul><ul><li>Is this a function? (Remember every x must map to...
<ul><li>Look at these graphs.  </li></ul><ul><li>Is it a function? </li></ul><ul><li>How can you tell if each x value maps...
<ul><li>Vertical line test </li></ul><ul><li>A method to test if there exists one and only one y value for every x value. ...
<ul><li>Is the graph a function? Why or why not? </li></ul>B A
Notice that it is OK if two x-values map to the same y-value. However the test has to work for every value of x.
<ul><li>Beware of the behavior of the graph in the extremes. It might look like it is becoming vertical, but it may just b...
<ul><li>How to determine if a relation is a function algebraically </li></ul><ul><ul><ul><ul><ul><li>Substitute y for f(x)...
<ul><li>How to determine if a relation is a function algebraically </li></ul><ul><ul><ul><ul><ul><li>Substitute y for f(x)...
<ul><li>Look at these familiar functions </li></ul><ul><li>f(x)=x 2  and  f(x)=x 3 </li></ul><ul><li>  y=x 2    y=x 3 </li...
<ul><li>How about x = y 2  ? </li></ul><ul><li>Solve for y  </li></ul><ul><li>y = ±√ x </li></ul><ul><li>Is this a functio...
<ul><li>Note that we need to pay attention to the domain of the function f….. </li></ul><ul><li>Consider f(x) = 2x+1/x-1 <...
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Functions

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Transcript of "Functions"

  1. 1. Functions By Adrienne Calomino
  2. 2. <ul><li>Objectives: </li></ul><ul><li>to identify the domain and range of a relation - review; </li></ul><ul><li>to recognize from a table of values or from coordinate points if a relation is a function; </li></ul><ul><li>to recognize from the graph of a familiar relations whether it is a function; </li></ul><ul><li>to use the vertical line test to determine if a graph is that of a function; </li></ul><ul><li>to recognize algebraically if a relation is a function. </li></ul>
  3. 3. x or input value y or output value <ul><li>Review domain and range of a relation </li></ul><ul><ul><li>Warm–up exercises </li></ul></ul>
  4. 4. <ul><li>When is a relation a function? </li></ul><ul><li>When each element in the domain maps to one and only one element in the range. </li></ul><ul><li>Does each element of the domain map to only one element of the range below? </li></ul>Dog Cat Duck Lion Pig Rabbit 11 10 7 Domain Range
  5. 5. <ul><li>Is this a Function? Why or why not? </li></ul>Dog Cat Duck Lion Pig Rabbit 11 10 7 Domain Range
  6. 6. <ul><li>How about this? </li></ul>Dog Cat Duck Lion Pig Rabbit 11 9 10 6 7 5 Domain Range <ul><li>This is a particular type of function we will see again later. </li></ul>
  7. 7. Function <ul><ul><li>A function maps each element of the domain to </li></ul></ul><ul><ul><li>one and only one element of the range </li></ul></ul>
  8. 8. Which is a function? Why? Table A Table B Table A Range Domain 7 5 6 4 5 4 5 3 3 2 3 1 2 1 output input Range Domain 8 7 7 6 5 5 5 4 3 3 3 2 2 1 output input
  9. 9. Which is a function? Why? Table A Table B Table A 120 Russian 95 Japan 48 Pakistan 48 France 72 Italy 72 Germany 47 Mexico 61 U.K. 438 China 219 U.S. Million Users Cell phone use by country Russian 120 Japan 95 Pakistan 48 France 48 Italy 72 Germany 72 Mexico 47 U.K. 61 China 438 U.S. 219 Cell phone use by country Million Users
  10. 10. <ul><li>Try coordinate points or ordered pairs </li></ul>Which is a function? Why? { (0,1) (0,-1), (1,2), (2,1), (3,1), (4,1) } { (0,1) (1,-1), (2,2), (-2,1), (3,2), (4,1) } Set A Set B State the domain and range.
  11. 11. <ul><li>1) Now try some from your worksheet </li></ul><ul><li>2) Group activity </li></ul><ul><ul><li>Make up two problems. </li></ul></ul><ul><ul><li>Each can be a table or set of coordinate points. </li></ul></ul><ul><ul><li>One should be a relation but not a function. </li></ul></ul><ul><ul><li>The other should be a function. </li></ul></ul><ul><ul><li>Do not include the answers. </li></ul></ul><ul><ul><li>Give the problems to the others in your group and see if they get it correct. </li></ul></ul><ul><ul><li>If not, explain the correct answer to them. </li></ul></ul>
  12. 12. <ul><li>Definition </li></ul><ul><li>We say “function”. We write f. </li></ul><ul><li>We say “f of x”. We write f(x). </li></ul><ul><li>Where x is a value in the domain and </li></ul><ul><li>f(x) is a value in the range. </li></ul><ul><li>f is the name of the function.   </li></ul><ul><li>x , which is placed within the parentheses, is called the argument of the function.   </li></ul><ul><li>It is upon the argument,x, that the function called f will &quot;operate.&quot; </li></ul>
  13. 13. Mapping a function
  14. 14. Why are these functions? Give an example of an argument of a function. What is the domain and range? Table A Table B f(0) f(-3) f(-2) f(-1) f(3) f(2) f(1) f(x) = 2x Range Domain 0 0 -6 -3 -4 -2 -2 -1 6 3 4 2 2 1 output or f(x) input or x f(0) f(-4) f(4) f(-2) f(2) f(-1) f(1) f(x) = x 2 Range Domain 0 0 16 -4 16 4 4 -2 4 2 1 -1 1 1 output or f(x) input or x
  15. 15. <ul><li>Now let’s relate coordinate points to a graph. </li></ul><ul><li>Is this a function? (Remember every x must map to one and only one y!) </li></ul>
  16. 16. <ul><li>Look at these graphs. </li></ul><ul><li>Is it a function? </li></ul><ul><li>How can you tell if each x value maps to only one y value? </li></ul>
  17. 17. <ul><li>Vertical line test </li></ul><ul><li>A method to test if there exists one and only one y value for every x value. </li></ul><ul><li>You draw vertical lines. </li></ul><ul><li>If the vertical line crosses the graph only once , it passes the test. </li></ul><ul><li>Why does this work? </li></ul>
  18. 18. <ul><li>Is the graph a function? Why or why not? </li></ul>B A
  19. 19. Notice that it is OK if two x-values map to the same y-value. However the test has to work for every value of x.
  20. 20. <ul><li>Beware of the behavior of the graph in the extremes. It might look like it is becoming vertical, but it may just be increasing or decreasing very gradually. </li></ul>
  21. 21. <ul><li>How to determine if a relation is a function algebraically </li></ul><ul><ul><ul><ul><ul><li>Substitute y for f(x) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Solve for y (if needed) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Is there only one y value in the range for each x value in the domain? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Try f(x)= 2x + 3 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li> y = 2x + 3 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>What type of relation is this? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Is it a function? </li></ul></ul></ul></ul></ul>
  22. 22. <ul><li>How to determine if a relation is a function algebraically </li></ul><ul><ul><ul><ul><ul><li>Substitute y for f(x) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Solve for y (if needed) </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Is there only one y value in the range for each x value in the domain? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Try f(x)= 2x 2 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li> y = 2x 2 </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>What type of relation is this? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Is it a function? </li></ul></ul></ul></ul></ul>
  23. 23. <ul><li>Look at these familiar functions </li></ul><ul><li>f(x)=x 2 and f(x)=x 3 </li></ul><ul><li> y=x 2 y=x 3 </li></ul>
  24. 24. <ul><li>How about x = y 2 ? </li></ul><ul><li>Solve for y </li></ul><ul><li>y = ±√ x </li></ul><ul><li>Is this a function? </li></ul><ul><li>Why or why not? </li></ul>
  25. 25. <ul><li>Note that we need to pay attention to the domain of the function f….. </li></ul><ul><li>Consider f(x) = 2x+1/x-1 </li></ul><ul><ul><ul><ul><li>y = 2x+1/ x-1 </li></ul></ul></ul></ul><ul><ul><li>How is the domain is restricted . Why? </li></ul></ul><ul><ul><ul><ul><ul><li>{ x| x ≠ 1 } </li></ul></ul></ul></ul></ul><ul><ul><li>f(x) = x+ 1/ x-2 </li></ul></ul><ul><ul><li>How is the domain is restricted. Why? </li></ul></ul><ul><ul><ul><ul><ul><li>{ x| x ≠ 2 } </li></ul></ul></ul></ul></ul>
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