functions

function

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