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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 β 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 />π¦=π(π₯) is a mathematical expression that is read as βy equals f of xβ. <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.<br />Β <br />
9. 9. Problem #8, pg. 19<br />(a) Finding πβ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β1=5.<br />Β <br />
12. 12. Pg. 19 #8 part(b) (contd.)<br />Part(b): To find πβ1 from the table, we simply find β1 in the x-column and look at the corresponding y-value.<br />Thus, for this function πβ1=β8.<br />Β <br />
13. 13. Pg. 19 #8 part (c ) (contd.)<br />Part (c ): Given ππ₯=π₯2+3π₯+8Β Β , we find π(β1) by substituting β1 for π₯ in the entire formula. So, we have <br />πβ1=β12+3β1+8 =1β3+8=6<br />So, Β Β πβ1=6.<br />Β <br />
14. 14. Pg. 21 #30<br />Find the domain of π¦=2π₯β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₯β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,β in interval notation.<br />Β <br />
15. 15. Pg. 21 #32<br />Find the domain of π¦=4+82π₯β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₯β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 ββ,Β 3βͺ3,β.<br />Β <br />
16. 16. Pg. 21 #33<br />Does π₯2+π¦2=4 describe π¦ as a function of π₯?<br />Solution: We need to determine if π¦=ππ₯.<br />Solve for the output variable, which is π¦.<br />Β π₯2+π¦2=4Β <br />βπ₯2Β Β Β Β Β Β Β Β Β Β βπ₯2<br />π¦2=4βπ₯2<br />Taking the square root on both sides,<br />π¦2 =Β±4βπ₯2<br />π¦=Β±4β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 />πΆπ₯=4000+12π₯ where x is the number of hats produced.<br />What is πΆ200? Interpret.<br />Sol: πΆ200=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: πΆ2500=4000+122500=\$34000<br />Β <br />