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function
is a rule that assigns to each element x in set A and exactly one element f(x) in set B
domain
is the set of all real numbers variable x can take such that the expression defining the function is real.?
range
the set of all values that the function takes when x takes values in the domain.
codomain
What can possibly come out of a function...usually the f(x) set
4 ways to describe a function
-verbally (by describing in words) -numerically (by table values) -visually (by a graph) -algebraically (by an explicit formula)
Vertical line test
A curve in the xy-plane is a function if no vertical line intersects the curve more than once
Piecewise Defined Functions
breaking up a non-function that does not pass the vertical line test into different pieces to evaluate as a function.
Absolute value of a number
The distance from zero
|x|= x if x≥0
       -x if x<0
Even function of f
if f(-x)=f(x) for all x in the domain
Odd function of f
if f(-x)= -f(x)
a function is increasing on an intercal I if
f(x)<f(y) whenever x<y with x and y in I
A function is decreasing on an interval I if
f(x)>f(y) if x<y with x and y in I
Linear Models
f(x)=mx+b
Polynomials
p(x)=anXn+an-1xn-1+......
Power functions
Case 1: a=n where n is a positive integer

Case 2: a=1/n where n is an integer
Trigonometric Functions
f(x)=sinx and g(x)=cos x
Rational Functions
f(x)=p(x) / q(x)

find asetopes by finding q(x)=0
sin-1x
domain - -1 to 1
range - Π/2 to -Π/2
cos-1x
domain -1 to 1
range 0 to Π
tan-1x
domain -∞ to ∞
range Π/2 to -Π/2
All Polynomials with degree of 2 are
Parabolas
number of turns in a polynomial graph
determines the degree of the polynomial
degree of a polynomial
the largest expontent in the formula
No derivative
a cusp
A function is continuous
at x=a if and only if lim f(x)=f(a) as x approaches a
to be continuous f(a) is
defined
To be continuous the limit of f(x) exists
as x approaches a
to be continuous lim f(x)=f(a) as
x approaches a
types of discontinuities
a) jump
b) removable
c) infinite
a function is continuous on an interval iff it is
continuous at each point in the interval
if f and g are continuous, so are
f+g, f-g, cf, fg, and f/g (except when a is not = 0)
Because of direct substitution
  • any polynomial is continuous everywhere
  • any reational function is continuous on it's domain
all the folowing are continuous on their domain

  • polynomial

  • rational

  • root function

  • trig functions

  • inverse trigonometric

  • exponential functions

  • logarithmic functions

if f is continuous at b
and lim g(x)=b as x approaches a, then lim(g(x))=f(limg(x)) as x approaches a
if g is continuous at a and f is continuous at g(a) then
fºg(x)=f(g(x)) is continuous at x=a
Intermediate value theorem
suppose that f is continuous on the closed interval [a,b] and let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) for which f(c)=N
lim f(x)=L as
x approaches infinity or minus infinity
horizontal asymptote
y=L
lim f(x)= + or - infinity as
x approaches a (from right and left)
vertical asymptote
a
limf(x)= + or - infinity as
x approaches + or - infinity
graph is
unbounded
a>0 exponential functions
decreasing domain all real range all real positive x axis is asetope
a>1 exponential function
increasing domain all real range all real positive x axis is asetope
logarithmic functions
inverse of exponential functions domain all real. positive range all real y axis is asetope
y=f(x) +c
shift graph up c units
y=f(x) -c
shifts graph c units down
y=f(x-c)
shifts graph c units to the right
y=f(x+c)
shifts graph c units to left
y=c f(x)
stretch the graph vertically by a factor of c
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