functions

function

* a rule that assigns every x-value to exactly one y-value
* passes “vertical line test”

domain

* all possible x-values
* input, dependent variable
* use interval notation

range

* all possible y-values
* output, dependent variable
* use interval notation

continuity

* no breaks in the graph
* a “continuous” curve

removable discontinuity

* the graph can be repaired by filling in a single point
* hole(s) in the graph

jump discontinuity

* a break in the graph
* if you trace the graph, you would have to jump to the next point

infinite discontinuity

* two or more pieces of the graph approach positive or negative infinity
* there is a vertical asymptote

increasing interval

* as x goes up, y goes up
* positive slope
* use x-values for the interval

decreasing interval

* as x goes up, y goes down
* negative slope
* use x-values for the interval

constant interval

* as x goes up, y stays the same
* 0 slope
* use x-values for the interval

relative/local maximum

* point where the graph changes from inc. to dec.
* could be more than one

relative/local minimum

* point where the graph changes from dec. to inc.
* could be more than one

absolute maximum

* highest point on the graph
* only one
* if positive or negative infinity, none

absolute minimum

* lowest point on the graph
* only one
* if pos. or neg. infinity, none

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

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

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 -∞

bounded below

* there is some number b that is less than or equal to every number in the range
* have an absolute minimum

bounded above

* there is some number b that is greater than or equal to every number in the range
* have an absolute maximum

bounded

when the graph is bounded above and below

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

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

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

indentity/linear

f(x)=x

\

quadratic

f(x)=x^2

cubic

f(x)=x^3

reciprocal

f(x)=1/x

square root

f(x)=\\|x

exponential

f(x)=e^x

logarithmic

f(x)=lnx

absolute value

f(x)=|x|

sine

f(x)=sinx

cosine

f(x)=cosx

greatest integer

f(x)=\[x\]

logistic

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

1/x^2

3\\|x