pre calculus functions and graphs. functions a function is a relation where each element of the...
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Pre Calculus
Functions and Graphs
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Functions
• A function is a relation where each element of the domain is paired with exactly one element of the range
• independent variable - x• dependent variable - y• domain - set of all values taken by
independent variable• range - set of all values taken by
the dependent variable
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Mapping
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Representing Functions• notation - f(x)• numerical model - table/list of
ordered pairs, matching input (x) with output (y)
• US Prison Polulation (thousands)Year Total Male Female
1980 329 316 13
1985 502 479 23
1990 774 730 44
1995 1125 1057 68
2000 1391 1298 93
2005 1526 1418 108
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• graphical model - points on a graph; input (x) on horizontal axis … output (y) on vertical
• algebraic model - an equation in two variables
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Vertical Line Test
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Finding the range
• implied domain - set of all real numbers for which expression is defined
• example: Find the range 31
yx
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31
yx
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Continuity
• http://www.calculus-help.com/tutorials
• function is continuous if you can trace it with your pencil and not lift the pencil off the paper
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Discontinuities
• point discontinuity– graph has a “hole”– called removable – example
2 3 4
4x x
f xx
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• jump discontinuity - gap between functions is a piecewise function
• example 4, 2
1 , 2
x xf x
x
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• infinite discontinuity - there is a vertical asymptote somewhere on the graph
• example 2
2
2 312
x xf x
x x
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Finding discontinuities
• factor; find where function undefined
• sub. each value back into original f(x)
• results …
# infinite disc.
0
0 point disc.
0
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Increasing - Decreasing Functions
• function increasing on interval if, for any two points
• decreasing on interval if
• constant on interval if
1 2 1 2 and , x x f x f x
1 2f x f x
1 2f x f x
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Example:
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22f x x
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Example:
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2 1x
g xx
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Boundedness of a Function
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Extremes of a Function
• local maximum - of a function is a value f(c) that is greater than all y-values on some interval containing point c.
• If f(c) is greater than all range values, then f(c) is called the absolute maximum
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• local minimum - of a function is a value f(c) that is less than all y-values on some interval containing point c.
• If f(c) is less than all range values, then f(c) is called the absolute minimum
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A
B
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local maxima
Absolute maximum
Absolute
minimumlocal minima
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Example: Identify whether the function has any local maxima
or minima
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4 27 6f x x x x
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Symmetry• graph looks same to left and right
of some dividing line• can be shown graphically,
numerically, and algebraically
• graph: 2f x x
x f(x)
-3 9
-1 1
0 0
1 1
3 9
numerically
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algebraically
• even function– symmetric about the y-axix– example
f x f x
22 8f x x
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• odd function– symmetric about the origin– example
f x f x
3 2f x x x
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Additional examples: even / odd
2 4 5 3 2
3 6
2
3 1 2
f x x x y x x x
g x x f x x
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Asymptotes
• horizontal - any horizontal line the graph gets closer and closer to but not touch
• vertical - any vertical line(s) the graph gets closer and closer to but not touch
• Find vertical asymptote by setting denominator equal to zero and solving
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End Behavior
• A function will ultimately behave as follows:– polynomial … term with the highest
degree– rational function … f(x)/g(x) take
highest degree in num. and highest degree in denom. and reduce those terms
– example
4 3
5 2
5 7 8 16 2 5x x x
f xx x