functions

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37 Terms

1

function

  • a rule that assigns every x-value to exactly one y-value

  • passes “vertical line test”

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2

domain

  • all possible x-values

  • input, dependent variable

  • use interval notation

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3

range

  • all possible y-values

  • output, dependent variable

  • use interval notation

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4

continuity

  • no breaks in the graph

  • a “continuous” curve

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5

removable discontinuity

  • the graph can be repaired by filling in a single point

  • hole(s) in the graph

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6

jump discontinuity

  • a break in the graph

  • if you trace the graph, you would have to jump to the next point

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7

infinite discontinuity

  • two or more pieces of the graph approach positive or negative infinity

  • there is a vertical asymptote

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8

increasing interval

  • as x goes up, y goes up

  • positive slope

  • use x-values for the interval

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9

decreasing interval

  • as x goes up, y goes down

  • negative slope

  • use x-values for the interval

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10

constant interval

  • as x goes up, y stays the same

  • 0 slope

  • use x-values for the interval

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11

relative/local maximum

  • point where the graph changes from inc. to dec.

  • could be more than one

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12

relative/local minimum

  • point where the graph changes from dec. to inc.

  • could be more than one

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13

absolute maximum

  • highest point on the graph

  • only one

  • if positive or negative infinity, none

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14

absolute minimum

  • lowest point on the graph

  • only one

  • if pos. or neg. infinity, none

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15

vertical asymptote

  • x=a is a va if f(x)→∞ or if f(x)→∞

  • to find, set the denominator of a function’s equation equal to zero

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16

horizontal asymptote

  • y=b is a ha if f(x)→b as x→∞ or as x→-∞

  • graphically, see where the ends of the graph go

  • algebraically, if the degree of the denominator is greater than the degree of the numerator, ha: y=0

    • if the degree of the denominator is less than the degree of the numerator, ha: none

    • if the degrees are equal, y=quotient of leading coefficients

  • the degree of a function is the highest exponent

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17

end behavior

  • the behavior at the left and right ends of the graph

  • as x→-∞, what does y approach?

  • as x→∞, what does y approach?

  • right eb: approaches ∞

  • left eb: approaches -∞

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18

bounded below

  • there is some number b that is less than or equal to every number in the range

    • have an absolute minimum

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19

bounded above

  • there is some number b that is greater than or equal to every number in the range

    • have an absolute maximum

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20

bounded

when the graph is bounded above and below

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21

even functions

  • y-axis symmetry

  • for each point (x,y) on the graph, the point (-x,y) is on the graph

  • f(-x)=f(x)

  • all graphs with y-axis symmetry are even functions

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22

not functions

  • for each point (x,y) on the graph, the point (x,-y) is on the graph

  • all graphs with x-axis symmetry are not functions

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23

odd functions

  • for each point (x,y) on the graph, the point (-x,-y) is on the graph

  • f(x)=-f(x)

  • all graphs with y-axis symmetry are odd functions

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24

indentity/linear

f(x)=x

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25

quadratic

f(x)=x^2

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26

cubic

f(x)=x^3

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27

reciprocal

f(x)=1/x

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28

square root

f(x)=\|x

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29

exponential

f(x)=e^x

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30

logarithmic

f(x)=lnx

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31

absolute value

f(x)=|x|

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32

sine

f(x)=sinx

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33

cosine

f(x)=cosx

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34

greatest integer

f(x)=[x]

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35

logistic

f(x)=1/1+e^-x

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36

1/x^2

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37

3\|x

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