Section 1.1Functions and Models<br />
Definitions<br />A Function is a rule that assigns to each input exactly one output.<br />The set of all inputs is called ...
Forms of functions.<br />Functions can be expressed in many forms.<br />Ordered pairs.<br />Arrow Diagrams.<br />Tables.<b...
Finding Domain and Range.<br />If the function is expressed as an ordered pair, arrow diagram or table then<br />List of a...
Finding Domain and Range (contd.)<br />Equations-(Refer Example 2, pg. 12)<br />If you have a variable in the denominator,...
Recognizing Functions.<br />If you have, <br />Tables or Co-ordinate pairs,<br />Check for duplicates in the input positio...
Recognizing Functions (contd.)<br />Graphs<br />Use the Vertical Line Test – A vertical line drawn anywhere should cross t...
Function notation.<br />𝑦=𝑓(π‘₯) is a mathematical expression that is read as β€œy equals f of x”.  <br />x is the input  also...
Problem #8, pg. 19<br />(a) Finding π‘“βˆ’1,<br />Find -1 on the x-axis:<br />Β <br />-1<br />
Pg. 19 #8 part(a) (contd.)<br />Go up to reach the curve that is the function. (On some other graphs, you might have to go...
Pg. 19 #8 part(a) (contd.)<br />Find the corresponding y-co-ordinate for that point on the curve.<br />The corresponding y...
Pg. 19 #8 part(b) (contd.)<br />Part(b): To find π‘“βˆ’1 from the table, we simply find βˆ’1 in the x-column and look at the cor...
Pg. 19 #8 part (c ) (contd.)<br />Part (c ): Given 𝑓π‘₯=π‘₯2+3π‘₯+8Β Β , we find 𝑓(βˆ’1) by substituting βˆ’1 for π‘₯ in the entire form...
Pg. 21 #30<br />Find the domain of 𝑦=2π‘₯βˆ’8.<br />Solution: To find the domain, we need to ensure that no input will lead us...
Pg. 21 #32<br />Find the domain of 𝑦=4+82π‘₯βˆ’6<br />Solution: To find the domain, we need to ensure that we do not include a...
Pg. 21 #33<br />Does π‘₯2+𝑦2=4 describe 𝑦 as a function of π‘₯?<br />Solution: We need to determine if 𝑦=𝑓π‘₯.<br />Solve for th...
Pg. 26 #60<br />The cost from the production of specialty golf hats is given by the function <br />𝐢π‘₯=4000+12π‘₯ where x is ...
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Section 1.1

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Section 1.1

  1. 1. Section 1.1Functions and Models<br />
  2. 2. Definitions<br />A Function is a rule that assigns to each input exactly one output.<br />The set of all inputs is called the domain.<br />The set of all outputs is called the range.<br />
  3. 3. Forms of functions.<br />Functions can be expressed in many forms.<br />Ordered pairs.<br />Arrow Diagrams.<br />Tables.<br />Graphs.<br />Equations.<br />Verbal descriptions.<br />
  4. 4. Finding Domain and Range.<br />If the function is expressed as an ordered pair, arrow diagram or table then<br />List of all the inputs is the domain.<br />List of all the outputs is the range.<br />If the function is expressed as graph, then visually determine the intervals or points on the input axis where the graph exists. The intervals might be open or closed.<br />
  5. 5. Finding Domain and Range (contd.)<br />Equations-(Refer Example 2, pg. 12)<br />If you have a variable in the denominator, then set the denominator =0 and solve to find the values NOT in the domain.<br />If you have a variable inside an even root set the expression β‰₯ 0 to find the domain.<br />Β <br />
  6. 6. Recognizing Functions.<br />If you have, <br />Tables or Co-ordinate pairs,<br />Check for duplicates in the input position.<br />Arrow diagrams,<br />Check for duplicates in the input, or,<br />An input with arrows assigning it to two different outputs.<br />
  7. 7. Recognizing Functions (contd.)<br />Graphs<br />Use the Vertical Line Test – A vertical line drawn anywhere should cross the graph exactly once.<br />Equations<br />Solve the equation for the output variable (usually y), if you have an expression which will give you 2 answers, (usually because of a Β±), the equation is not a function.<br />Β <br />
  8. 8. Function notation.<br />𝑦=𝑓(π‘₯) is a mathematical expression that is read as β€œy equals f of x”. <br />x is the input also called the independent variable.<br />y is the output also called the dependent variable.<br />f is the name of the function.<br />f(a) is notation for the function evaluated for an input value of π‘₯=π‘Ž.<br />Β <br />
  9. 9. Problem #8, pg. 19<br />(a) Finding π‘“βˆ’1,<br />Find -1 on the x-axis:<br />Β <br />-1<br />
  10. 10. Pg. 19 #8 part(a) (contd.)<br />Go up to reach the curve that is the function. (On some other graphs, you might have to go down.)<br />-1<br />
  11. 11. Pg. 19 #8 part(a) (contd.)<br />Find the corresponding y-co-ordinate for that point on the curve.<br />The corresponding y co-ordinate is 5. Therefore π‘“βˆ’1=5.<br />Β <br />
  12. 12. Pg. 19 #8 part(b) (contd.)<br />Part(b): To find π‘“βˆ’1 from the table, we simply find βˆ’1 in the x-column and look at the corresponding y-value.<br />Thus, for this function π‘“βˆ’1=βˆ’8.<br />Β <br />
  13. 13. Pg. 19 #8 part (c ) (contd.)<br />Part (c ): Given 𝑓π‘₯=π‘₯2+3π‘₯+8Β Β , we find 𝑓(βˆ’1) by substituting βˆ’1 for π‘₯ in the entire formula. So, we have <br />π‘“βˆ’1=βˆ’12+3βˆ’1+8 =1βˆ’3+8=6<br />So, Β Β π‘“βˆ’1=6.<br />Β <br />
  14. 14. Pg. 21 #30<br />Find the domain of 𝑦=2π‘₯βˆ’8.<br />Solution: To find the domain, we need to ensure that no input will lead us to take the square root of a negative number. So, we need,<br />2π‘₯βˆ’8Β β‰₯0<br />+8Β Β Β Β Β Β Β Β Β Β +8Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β <br />2π‘₯β‰₯8Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β <br />2π‘₯2β‰₯82Β Β Β ->Β Β Β Β π‘₯β‰₯4Β <br />Thus, our domain is π‘₯β‰₯4.Β  Or 4,∞ in interval notation.<br />Β <br />
  15. 15. Pg. 21 #32<br />Find the domain of 𝑦=4+82π‘₯βˆ’6<br />Solution: To find the domain, we need to ensure that we do not include an input that would cause the denominator to be zero. So, we need, <br />2π‘₯βˆ’6β‰ 0<br />Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β +6Β Β Β Β Β Β Β +6<br />2π‘₯β‰ 6<br />2π‘₯2β‰ 62<br />π‘₯β‰ 3<br />So, our domain should exclude 3. In interval notation, this would be βˆ’βˆž,Β 3βˆͺ3,∞.<br />Β <br />
  16. 16. Pg. 21 #33<br />Does π‘₯2+𝑦2=4 describe 𝑦 as a function of π‘₯?<br />Solution: We need to determine if 𝑦=𝑓π‘₯.<br />Solve for the output variable, which is 𝑦.<br />Β π‘₯2+𝑦2=4Β <br />βˆ’π‘₯2Β Β Β Β Β Β Β Β Β Β βˆ’π‘₯2<br />𝑦2=4βˆ’π‘₯2<br />Taking the square root on both sides,<br />𝑦2 =Β±4βˆ’π‘₯2<br />𝑦=Β±4βˆ’π‘₯2<br />So, y is not a function of x.<br />Β <br />
  17. 17. Pg. 26 #60<br />The cost from the production of specialty golf hats is given by the function <br />𝐢π‘₯=4000+12π‘₯ where x is the number of hats produced.<br />What is 𝐢200? Interpret.<br />Sol: 𝐢200=4000+12200=6400<br />The cost of producing 200 hats is $6400.<br />(b) What is the cost from the production of 2500 hats? Write in function notation.<br />Sol: 𝐢2500=4000+122500=$34000<br />Β <br />

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